Combinatorics of Acyclic Orientations of Graphs:
Algebra, Geometry and Probability
by
ARCHIVES
ISTITI.TE
k1ASSACHFe',TTF
OF k "'HNOLOLGY
Benjamin Iriarte Giraldo
M.A., San Francisco State University (2010)
B.S., Universidad de los Andes (2009)
JUN 3 0 2015
LIBRARIES
Submitted to the Department of Mathematics
in partial fulfillment of the requirements for the degree of
Doctor of Philosophy in Mathematics
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
June 2015
@ Massachusetts Institute of Technology 2015. All rights reserved.
Signature redacted
A u th o r ............ p ....
.
.........................................
Department of Mathematics
February 25, 2015
Certified by..
Signature redacted
V/
Richard P. Stanley
Professor of Applied Mathematics
Thesis Supervisor
Accepted by ...................
Signature redacted
lAfichel X. Goemans
Chairman, Department Committee on Graduate Theses
Combinatorics of Acyclic Orientations of Graphs: Algebra,
Geometry and Probability
by
Benjamin Iriarte Giraldo
Submitted to the Department of Mathematics
on February 25, 2015, in partial fulfillment of the
requirements for the degree of
Doctor of Philosophy in Mathematics
Abstract
This thesis studies aspects of the set of acyclic orientations of a simple undirected
graph. Acyclic orientations of a graph may be readily obtained from bijective labellings of its vertex-set with a totally ordered set, and they can be regarded as
partially ordered sets. We will study this connection between acyclic orientations of
a graph and the theory of linear extensions or topological sortings of a poset, from
both the points of view of poset theory and enumerative combinatorics, and of the
geometry of hyperplane arrangements and zonotopes. What can be said about the
distribution of acyclic orientations obtained from a uniformly random selection of
bijective labelling? What orientations are thence more probable? What can be said
about the case of random graphs? These questions will begin to be answered during
the first part of the thesis. Other types of labellings of the vertex-set, e.g. proper
colorings, may be used to obtain acyclic orientations of a graph, as well. Motivated
by our first results on bijective labellings, in the second part of the thesis, we will use
eigenvectors of the Laplacian matrix of a graph, in particular, those corresponding
to the largest eigenvalue, to label its vertex-set and to induce partial orientations of
its edge-set. What information about the graph can be gathered from these partial
orientations? Lastly, in the third part of the thesis, we will delve further into the
structure of acyclic orientations of a graph by enhancing our understanding of the
duality between the graphical zonotope and the graphical arrangement with the lens
of Alexander duality. This will take us to non-crossing trees, which arguably vastly
subsume the combinatorics of this geometric and algebraic duality. We will then
combine all of these tools to obtain probabilistic results about the number of acyclic
orientations of a random graph, and about the uniformly random choice of an acyclic
orientation of a graph, among others.
Thesis Supervisor: Richard P. Stanley
Title: Professor of Applied Mathematics
3
Acknowledgments
During my years at MIT, I have had the fortune to be supervised by the same person
whose mathematics once inspired me to become a mathematician in the first place.
I am thankful to Richard P. Stanley for the obvious reasons why a student should
be thankful to his/her advisor, but far more importantly, for his own mathematical
work, which not only has inspired me, but also a vast number of people in the world.
No mathematician on earth, at this stage of his/her career, can have the luxury
to not acknowledge the contributions to their own scientific development of a large
number of people.
To get to this point, people need other people who inspire them. This is specially
true in mathematics, where our job demands from us to regularly "steal" someone
else's beautiful ideas and use them to generate other new ideas. Some of these sets of
ideas that inspired me and that are present in this thesis, and their authors/divulgers,
need to be thanked for. I'd like to thank the following mathematicians: Richard
P. Stanley, Gian-Carlo Rota, Christos A. Athanasiadis, Bernd Sturmfels, Alexander Postnikov, Persi Diaconis, Ezra Miller, Tibor Gallai, Victor Guillemin, James
Munkres, Arthur Engel, Roger Heath-Brown, Colin McDiarmid, Bela Bollobas, JeanPierre Serre, Matthias Beck, C.F. Gauss and the compass, and Oscar Bernal.
People then need teachers who can help them bring their inspiration into action,
and this transition is crucial. In my case, special thanks must go to Federico Ardila,
Richard P. Stanley, Jacob Fox, Maricarmen Martinez, Jean Carlos Cortissoz, Serkan
Hosten, and the members of my thesis comittee: Michelle Wachs and Henry Cohn.
In order to make the best out of themselves, people also need colleagues who always
give the best and who always ask for the best. I need to thank here my own share
of excellent co-workers: Pablo Garcia, Pedro Rangel, Carlos Coelho, Brad Chase,
Alejandro H. Morales, Michael Donovan, Joel B. Lewis, Luis G. Serrano, Haotian
Pang, and Alex Fink.
Because there is always the risk of burnouts, people need to have good roots to
keep the hard work at safe levels. My sincere thanks must go here to my family, Valeria Rueda, Stefan Bbssinger, Juan Camilo Velasquez, Andr6s D. Jaramillo, Janeth
Velasco, Alejandro H. Morales, Pablo Garcia, Natalia Duque, Andr6s Cubillos, Alejandra Falla, and Pedro Rangel.
And even with all these weapons at hand, things might just go wrong and life can
get complicated. That's why people need to surround themselves with individuals of
"magical power". These are the ones that, due to some uncanny wisdom and knowhow, can change the course of your professional life in a couple of conversations.
Very special thanks to Federico Ardila, Andr6s Villaquiran, Sergey Fomin, Jacob
Fox, Bernd Sturmfels, and Carly Klivans.
Now, mathematics or any kind of work is a human effort and therefore, is bound
to our human condition; we humans need support, hope, and light. Special thanks
go to my mother, Maria Amparo Giraldo, whom I greatly admire.
Lastly, people need to have a reason why. My greatest thanks then go to my wife,
to my son, and to God.
4
5
Contents
Preface
10
1
2
3
Linear Extensions.
1.1 Introduction. . . . . . . . . . . . . . . . . . . .
1.2 Introductory results. . . . . . . . . . . . . . . .
1.2.1 The case of bipartite graphs. . . . . . . .
1.2.2 O dd cycles. . . . . . . . . . . . . . . . .
1.3 Comparability graphs. . . . . . . . . . . . . . .
1.3.1 G eom etry. . . . . . . . . . . . . . . . . .
1.3.2 Poset theory. . . . . . . . . . . . . . . .
1.4 Beyond comparability and enumerative results .
1.4.1 A useful technique. . . . . . . . . . . . .
1.4.2 General bounds for the main statistic. .
1.4.3 Random graphs . . . . . . . . . . . . . .
1.5 Further techniques. . . . . . . . . . . . . . . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
13
13
15
15
17
19
20
22
24
25
25
27
29
Largest Eigenvalue of the Laplacian Matrix.
2.1 Introduction. . . . . . . . . . . . . . . . . . .
2.2 Background and definitions. . . . . . . . . . .
2.2.1 The graphical arrangement. . . . . . .
2.2.2 Modular decomposition. . . . . . . . .
2.2.3 Comparability graphs. . . . . . . . . .
2.2.4 Linear algebra. . . . . . . . . . . . . .
2.2.5 Spectral theory of the Laplacian. . . .
2.3 Largest Eigenvalue of a Comparability Graph.
2.4 A characterization of comparability graphs . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
33
33
35
35
36
37
38
39
39
49
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
-
. .
. .
. .
. .
. .
. .
-. .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
Spanning Trees.
3.1 Introduction. . . . . . . . . . . . . . . . . .
3.2 Polytopal complexes for acyclic orientations.
3.2.1 A Classical Polytope. . . . . . . . . .
3.2.2 One More Degree of Freedom. . . . .
3.3 Two ideals for acyclic orientations. . . . . .
3.4 Non-crossing trees. . . . . . . . . . . . . . .
3.4.1 Standard monomials of TG . . . . . . .
6
.
.
.
.
.
.
. . . . .
. . .
51
51
53
53
59
67
73
77
3.5
3.4.2 Non-crossing partitions. . . . . . . . . . . . . .
A pplications. . . . . . . . . . . . . . . . . . . . . . . .
3.5.1
Random Acyclic Orientations of a Simple Graph:
3.5.2 Acyclic Orientations of a Random Graph. . . .
3.5.3 k-Neighbor Bootstrap Percolation. . . . . . . .
7
. . . . . . . .
. . . . . . . .
Markov Chains.
. . . . . . . .
. . . . . . . .
82
85
85
92
94
List of Figures
1-1
1-2
2-1
3-1
3-2
3-3
3-4
3-5
3-6
3-7
An example of the function E for the case of bipartite graphs. Squares
show the numbers that will be flipped at each step, and dashed arrows
indicate arrows whose orientation still needs to be reversed. . . . . . .
An example of the function 8' for the case of odd cycles. Squares show
the numbers that will be flipped at each step. Dashed arrows indicate
arrows whose orientation still needs to be reversed, while dashed-dotted
arrows indicate those whose orientation will never be reversed. In
particular, 4 will remain labeling the same vertex during all steps. . .
(2-1a) Hasse diagram of a poset P on ground set [8]. (2-1b) Comparability graph G = G([8], E) of the poset P, where the closed regions depict the maximal proper modules of G. (2-1c) Unit eigenvector x E EAmax of G fully calculated, where dim (Ea)
= 1. Arrows
represent the induced orientation Ox of G. Notice the relation between
. . . . . . . . . . . . . . . .
OX, the modules of G, and the poset P.
Examples of p.a.o.'s and the order relation i of Definition 3.2.11. . .
Visual aids/guides to the proofs of (3-2a) Proposition 3.2.23 and (3-2b)
Proposition 3.2.20.1. (3-2a) also offers an example for Definition 3.2.21.
An example to the proof of Claim i.2 in Theorem 3.2.25. . . . . . .
Example of a planar depiction, according to Definition 3.4.2. . . . . .
Fully worked out example illustrating the central dogma of Section 3.4.
Theorems 3.4.13 and 3.4.22 are dwelled on in tables 3-5e.i and 3-5e.ii,
respectively, and in particular, fa = P. . . .
. .
. . . . . .. .
Example of the bijection of Proposition 3.4.15. The selected spanning
tree of Gr (in red) corresponds to the acyclic orientation 0 of G presented.
Examples of Definitions 3.5.9 and 3.5.12 for the 4-cycle C4 . In 3-7a,
we present the 1-skeleton of the graphical zonotope of C4, a rhombic
dodecahedron, where the Cover-Reversal random walk runs; notably, it
is not a regular graph. If four diagonals are added to the graph as shown
in 3-7b, we obtain a 4-regular graph, AO"er in Proposition 3.5.14,
where the Interval-Reversal random walk runs. . . . . . . . . . . . . .
8
18
19
38
56
62
65
75
77
81
92
9
Preface
Two algorithmic problems have been in our mind since, approximately, the summer
of 2013. A simplified and more restrictive version of the first problem is the following:
1
Problem 1. Let n c P. Call a sequence d = (di, d 2 ,..., d,) E NM"
a graphical
sequence if there exists a simple undirected graph G = G([n], E) such that, for all
i E [n], the degree of vertex i in G is equal to di. In other words, d is a graphical
sequence if it is the degree sequence of a simple undirected graph on vertex-set [n].
Suppose now that d E N" is a graphical sequence. Then, can we find an "efficient"
and "transparent" algorithm to select, uniformly at random, a simple undirected graph
on vertex-set [n] with degree sequence d?
The second problem is the following:
Problem 2. Let n E P and let G = G([n], E) be a simple undirected graph. Call a
map 0 : E -+ [n] x [n] such that 0(e) E {(i, j), (j,i)} for all e := {i, j} E E, an acyclic
orientation of E or G, if the directed-graph on vertex-set [n] and directed-edge set
O[E] has no directed-cycles.
Generally, G possesses an exponential (on n) number of different acyclic orientations. But then, can we find an "efficient" and "transparent" algorithm to select,
uniformly at random, an acyclic orientation of G?
Perhaps, the reader might argue, these problems may be solvable in practice today
by masterfully combining difficult tools from modern mathematical technology. We,
reluctantly, would tend to be dissatisfied with these modern solutions and argue
that instead, they still remain short in discovering and exploiting the fundamental
properties of graphical sequences and/or of acyclic orientations and are, therefore,
prone to be suboptimal. By not requiring our solutions to these problems to be
"transparent", not only are we missing the opportunity to discover new mathematical
theories, but also the opportunity to make these discoveries useful to outsiders. An
inspiring example here to illustrate our point (to name at least one) are the surprising
results of Broder {1989}. In that paper, the author introduced yet another method to
select, uniformly at random, a spanning tree of a connected simple undirected graph.
This thesis is a by-product of our efforts to solve Problem 2, and it presents three
of the projects that we pursued in that vein. These three building blocks of the
manuscript are, however, of independent mathematical interest and as such, they will
be presented as self-contained units that are not only independent from Problem 2,
but also from each other.
During the first part, in Chapter 1, we will study the following problem:
10
Given an underlying undirected simple graph, consider the set of all acyclic
orientations of its edges. Each of these orientations induces a partial order on
the vertices of our graph, and we can count the number of linear extensions of
these posets. Then, what choice of acyclic orientation maximizes the number
of linear extensions of the corresponding poset?
This problem will be solved essentially for comparability graphs and odd cycles, presenting several proofs. The corresponding enumeration problem for arbitrary simple
graphs will be studied, including the case of random graphs; this will culminate in 1)
new bounds for the volume of the stable polytope and 2) strong concentration results
for our main statistic and for the graph entropy, which hold true a.s. for random
graphs. We will then argue that this problem springs up naturally in the theory of
graphical arrangements and graphical zonotopes.
In Chapter 2, we will study the eigenspace of the Laplacian matrix of a simple
graph corresponding to the largest eigenvalue, subsequently arriving at the theory of
modular decomposition of T. Gallai.
Lastly, in Chapter 3, we will introduce three polytopal cell complexes associated
with partial acyclic orientations of a simple graph, which generalize acyclic orientations. Using the theory of cellular resolutions, we will observe that two of these
polytopal cell complexes minimally resolve certain special combinatorial polynomial
ideals related to acyclic orientations. These ideals will explicitly be found to be
Alexander dual, which relative to comparable results in the literature, generalizes
in a cleaner and more illuminating way the well-known duality between permutohedron and tree ideals. The combinatorics underlying these results will then naturally
lead us to a canonical way to represent rooted spanning forests of a labelled simple
graph as non-crossing trees, and these representations will be observed to carry a
plethora of information about generalized tree ideals and acyclic orientations of a
graph, and about non-crossing partitions of a totally ordered set. We will then study
in detail a small sample of the enumerative and structural consequences of collecting
and organizing this information. Applications of this combinatorial miscellany will
then be introduced and explored, namely: Stochastic processes on state space equal
to the set of all acyclic orientations of a simple graph, including irreducible Markov
chains, which exhibit stationary distributions ranging from linear extensions-based to
uniform; a surprising formula for the expected number of acyclic orientations of an
Erdbs-Renyi random graph; and a purely algebraic presentation of the main problem
in bootstrap percolation, likely making it tractable to explore the set of all percolating
sets of a graph with a computer.
11
12
Chapter 1
Linear Extensions.
1.1
Introduction.
Linear extensions of partially ordered sets have been the object of much attention
and their uses and applications remain increasing. Their number is a fundamental
statistic of posets, and they relate to ever-recurring problems in computer science
due to their role in sorting problems. Still, many fundamental questions about linear
extensions are unsolved, including the well-known 1/3-2/3 Conjecture. Efficiently
enumerating linear extensions of certain posets is difficult, and the general problem
has been found to be OP-complete in Brightwell and Winkler {1991}.
Acyclic directed-graphs, and similarly, acyclic orientations of simple undirected
graphs, are closely related to posets, and their problem-modeling values in several
disciplines, including the biological sciences, needs no introduction. On this chapter,
we study the following problem:
Problem 1.1.1. Suppose that there are n individuals with a known contagious disease, and suppose that we know which pairs of these individuals were in the same
location at the same time. Assume that at some initial points, some of the individuals fell ill, and then they started infecting other people and so forth, spreading the
disease until all n of them were infected. Then, assuming no other knowledge of the
situation, what is the most likely way in which the disease spread out?
Suppose that we have an underlying connected undirected simple graph G =
G(V, E) with n vertices. If we first pick uniformly at random a bijection f : V -+
[n], and then orient the edges of E so that for every {u, v} E E we select (u, v)
(read u directed to v) whenever f(u) < f(v), we obtain an acyclic orientation of
E. In turn, each acyclic orientation induces a partial order on V in which u < v
if and only if there is a directed-path (u, u1 ), (ui, u2 ), .. . , (uk, v) in the orientation.
In general, several choices of f above will result in the same acyclic orientation.
However, the most likely acyclic orientations so obtained will be the ones whose
induced posets have the maximal number of linear extensions, among all posets arising
from acyclic orientations of E. Our problem then becomes that of deciding which
acyclic orientations of E attain this optimality property of maximizing the number of
13
linear extensions of induced posets. This problem, referred to throughout this chapter
as the main problem for G, was raised by Saito {2007} for the case of trees, yet, a
solution for the case of bipartite graphs had been obtained already by Stachowiak
{1988}. The main problem brings up the natural associated enumerative question:
For a graph G, what is the maximal number of linear extensions of a poset induced
by an acyclic orientation of G? This statistic for simple graphs will be herein referred
to as the main statistic (Definition 1.3.12).
The central goal of this initial chapter on the subject will be to begin a rigorous
study of the main problem from the points of view of structural and enumerative
combinatorics. We will introduce 1) techniques to find optimal orientations of graphs
that are provably correct for certain families of graphs, and 2) techniques to estimate
the main statistic for more general classes of graphs and to further understand aspects
of its distribution across all graphs.
In Section 1.2, we will present an elementary approach to the main problem for
both bipartite graphs and odd cycles. This will serve as motivation and preamble for
the remaining sections. In particular, in Section 1.2.1, a new solution to the main
problem for bipartite graphs will be obtained, different to that of Stachowiak {1988}
in that we explicitly construct a function that maps injectively linear extensions of
non-optimal acyclic orientations to linear extensions of an optimal orientation. As
we will observe, optimal orientations of bipartite graphs are precisely the bipartite
orientations (Definition 1.2.1). Then, in Section 1.2.2, we will extend our solution
for bipartite graphs to odd cycles, proving that optimal orientations of odd cycles are
precisely the almost bipartite orientations (Definition 1.2.7).
In Section 1.3, we will introduce two new techniques, one geometrical and the other
poset-theoretical, that lead to different solutions for the case of comparability graphs.
Optimal orientations of comparability graphs are precisely the transitive orientations
(Definition 1.3.2), a result that generalizes the solution for bipartite graphs. The
techniques developed on Section 1.3 will allow us to re-discover the solution for odd
cycles and to state inequalities for the general enumeration problem in Section 1.4.
The recurrences for the number of linear extensions of posets presented in Corollary 1.3.11 had been previously established in Edelman et al. {1989} using promotion
and evacuation theory, but we will obtain them independently as by-products of certain network flows in Hasse diagrams. Notably, Stachowiak {1988} had used some
instances of these recurrences to solve the main problem for bipartite graphs.
Further on, in Section 1.4, we will also consider the enumeration problem for the
case of random graphs with distribution G , 0 < p < 1, and obtain tight concentration results for our main statistic, across all graphs. Incidentally, this will lead to new
inequalities for the volume of the stable polytope and to a very strong concentration
result for the graph entropy (as defined in Csiszir et al. {1990}), which hold a.s. for
random graphs.
Lastly, in Section 1.5, we will show that the main problem for a graph arises
naturally from the corresponding graphical arrangement by asking for the regions
with maximal fractional volume (Proposition 1.5.2). More surprisingly, we will also
observe that the solutions to the main problem for comparability graphs and odd
cycles correspond to certain vertices of the corresponding graphical zonotopes (The-
14
orem 1.5.3).
Convention 1.1.2 (For Chapter 1). Let G = G(V, E) be a simple undirected graph.
Formally, an (complete) orientation 0 of E or of G is a map 0 : E -+ V 2 such that
for all e := {u, v} E E, we have 0(e) E {(u, v), (v, u)}. Furthermore, 0 is said to be
acyclic if the directed-graph on vertex-set V and directed-edge-set O[E] is acyclic. On
numerous occasions, we will somewhat abusively also identify an acyclic orientation
0 of E with the set O[E], or with the poset that it induces on V, doing this with the
aim to reduce extensive wording.
When defining posets herein, we will also try to make clear the distinction between
the ground set of the poset and its order relations.
1.2
1.2.1
Introductory results.
The case of bipartite graphs.
The goal of this section is to present a combinatorial proof that the number of linear
extensions of a bipartite graph G is maximized when we choose a bipartite orientation
for G. Our method is to find an injective function from the set of linear extensions of
any fixed acyclic orientation to the set of linear extensions of a bipartite orientation,
and then to show that this function is not surjective whenever the initial orientation
is not bipartite. Throughout the section, let G be bipartite with n > 1 vertices.
Definition 1.2.1. Suppose that G = G(V, E) has a bipartition V = V1 Li V2 . Then,
the orientations that either choose (v 1 , v 2 ) for all {v 1 , v 2 } E E with v 1 E V1 and
v 2 G V2 , or (v 2 7 v 1 ) for all {v 1 ,v 2 } E E with v 1 E V and v 2 E V2 , are called bipartite
orientations of G.
Definition 1.2.2. For a graph G on vertex-set V with |VI
Bij(V, [n]) the set of bijections from V to [n].
=
n, we will denote by
As a training example, we consider the case when we transform linear extensions
of one of the bipartite orientations into linear extensions of the other bipartite orientation. We expect to obtain a bijection for this case.
Proposition 1.2.3. Let G = G(V, E) be a simple connected undirected bipartite
graph, with n = |VI. Let Odown and 0,, be the two bipartite orientationsof G. Then,
there exists a bijection between the set of linear extensions of Odon and the set of
linear extensions of OU.
Proof. Consider the automorphism rev of the set Bij(V, [n]) given by rev(f)(v) = n+
1- f(v) for all v E V and f E Bij(V, [n]). It is clear that (rev o rev)(f) = f. However,
since f(u) > f(v) implies rev(f)(u) < rev(f)(v), then rev reverses all directed-paths
in any f-induced acyclic orientation of G, and in particular the restriction of rev to
the set of linear extensions of Odown has image O,,P, and viceversa.
15
We now proceed to study the case of general acyclic orientations of the edges of
G. Even though similar in flavour to Proposition 1.2.3, our new function will not in
general correspond to the function presented in the proposition when restricted to
the case of bipartite orientations.
To begin, we define the main automorphisms of Bij(V, [n]) that will serve as building blocks for constructing the new function.
Definition 1.2.4. Consider a simple graph G = G(V, E) with |VI = n. For different
vertices u, v E V, let revu, be the automorphism of Bij(V, [n]) given by the following
rule: For all f G BIj(V, [n]), let
reVUV (f)(u)
:reVU"(f)(V)
revUV(f)(w)
= f (v),
= f (U),
= f(w) if w
c
V\{u,v}.
It is clear that (revuv o revuv)(f) = f for all f E Bij(V, [n]). Moreover, we will
need the following technical observation about revuv.
Observation 1.2.5. Let G = G(V, E) be a simple graph with |VI = n and consider
a bijection f G Bij(V, [n]). Then, if for some u,v,x, y E V with f(u) < f(v) we have
that revu,(f)(x) > revec(f)(y) but f(x) < f(y), then f(u)
<;
f(x) < f(y) < f(v) and
furthermore, at least one of f(x) or f(y) must be equal to one of f(u) or f(v).
Let us present the main result of this section, obtained based on the interplay
between acyclic orientations and bijections in Bij(V, [n]).
Theorem 1.2.6. Let G = G(V, E) be a connected bipartite simple graph with VI = n,
and with bipartite orientations Odown and OUP. Let also 0 be an acyclic orientation
of G. Then, there exists an injective function E from the set of linear extensions of
o to the set of linear extensions of OU, and furthermore, 9 is surjective if and only
if O = Oup or 0 = Odown.
Proof. Let
f
be a linear extension of 0, and without loss of generality assume that
o # Oup. We seek to find a function 9 that transforms f into a linear extension
of OU, injectively. The idea will be to describe how 8 acts on f as a composition
of automorphisms of the kind presented in Definition 1.2.4. Now, we will find the
terms of the composition in an inductive way, where at each step we consider the
underlying configuration obtained after the previous steps. In particular, the choice
of terms in the composition will depend on f. The inductive steps will be indexed
using a positive integer variable k, starting from k = 1, and at each step we will know
an acyclic orientation Ok of G, a set Bk and a function fk. The set Bk C V will
always be defined as the set of all vertices incident to an edge whose orientation in Ok
and Op differs, and fk will be a particular linear extension of Ok that we will define.
Initially, we set 01 = 0 and fi = f, and calculate B 1 . Now, suppose that for
some fixed k > 1 we know Ok, Bk and fk, and we want to compute Ok+1, Bk+1 and
fk+1. If Bk = 0, then Ok = Oup and fk is a linear extension of Oup, so we stop our
recursive process. If not, then Bk contains elements uk and vk such that fk(uk) and
16
fk(vk) are respectively minimum and maximum elements of fk(Bk) 9 [n]. Moreover,
0
k+1 be the acyclic orientation of G
Uk # Vk. We will then let fk+1 := revukVk(fk),
induced by fk+1, and calculate Bk+1 from Ok+1If we let m be the minimal positive integer for which Bm+ = 0, then 8(f) =
(revumvm o ... o revU 2 V 2 o revulv1 ) (f). The existence of m follows from observing that
Bk+1 C Bk whenever Bk 74 0. In particular, if Bk # 0, then Uk, Vk E Bk\Bk+1 and
so 1 < m < [IB1IJ. It follows that the pairs {{'uk, Vk}ke[m
are pairwise disjoint,
for all k c [m], and f(ui) < f(u 2 ) < -
<
f(Um) < f(Vm) < ... < f(v 2 ) < f(vi). As a consequence, the automorphisms in the
composition description of 9 commute. Lastly, fm+1 will be a linear extension of O"
and we stop the inductive process by defining 8(f) = fm+i.
To prove that 9 is injective, note that given 0 and fm+i as above, we can recover
uniquely f by imitating our procedure to find 9(f). Firstly, set g, := fm+1 and Q, :=
OUP, and compute C1 C V as the set of vertices incident to an edge whose orientation
differs in Q, and 0. Assuming prior knowledge of Qk, Ck and gA, and whenever
Ck 74 0 for some positive integer k, find the elements of Ck whose images under A are
maximal and minimal in gk(Ck). By the discussion above and Observation 1.2.5, we
check that these are respectively and precisely Uk and Vk. Resembling the previous
case, we will then let gk+1 := revukVk(gk), Qk+1 be the acyclic orienation of G induced
by 9A, and compute Ck+1 accordingly as the set of vertices incident to an edge with
different orientation in Qk+1 and 0. Clearly gm+1 = f, and the procedure shows that
9 is invertible in its image.
To establish that 9 is not surjective whenever 0 74 Odown, note that then 0
contains a directed-2-path (w, u) and (u, v). Without loss of generality, we may
assume that the orientation of these edges in 0, is given by (w, u) and (v, u). But
then, a linear extension g of O,, in which g(u) = n and g(v) = 1 is not in Im(8)
since otherwise, using the notation and framework discussed above, there would exist
different i, j E [m] such that ui = u and vj = v, which then contradicts the choice of
u1 and v 1. This completes the proof.
f(uk) = fk(uk) and f(vk)
1.2.2
=
fk(vk)
Odd cycles.
In this section G = G(V, E) will be a cycle on 2n + 1 vertices with n > 1. The case
of odd cycles follows as an immediate extension of the case of bipartite graphs, but
it will also be covered under a different guise in Section 1.4. As expected, the acyclic
orientations of the edges of odd cycles that maximize the number of linear extensions
resemble as much as possible bipartite orientations. This is now made precise.
Definition 1.2.7. For an odd cycle G = G(V, E), we say that an ayclic orientationof
its edges is almost bipartite if under the orientation there exists exactly one directed2-path, i.e. only one instance of (u,v) and (v,w) in the orientation with u,v,w E V.
17
B, = { L 4,
f
78. 10.11
9
132
11
10)
1'2)
2
8
5
3
710
9
11
7
2
8
5
3
6
10
B={7.8}
1
= 14,7 ..
1
B-=
6
S
5
3
10
9
11
8
2
7
5
3
6
10
6
4
0
1
4
2
Figure 1-1: An example of the function 8 for the case of bipartite graphs. Squares
show the numbers that will be flipped at each step, and dashed arrows indicate arrows
whose orientation still needs to be reversed.
Theorem 1.2.8. Let G = G(V, E) be an odd cycle on 2n + 1 vertices with n > 1.
Then, the acyclic orientations of E that maximize the number of linear extensions
are the almost bipartite orientations.
First proof. Since the case when n = 1 is straightforward let us assume that n> 2
and consider an arbitrary acyclic orientation 0 of G. Again, our method will be to
construct an injective function e' that transforms every linear extension of 0 into a
linear extension of some fixed almost bipartite orientation of G, where the specific
choice of almost bipartite orientation will not matter by the symmetry of G.
To begin, note that there must exist a directed-2-path in 0, say (u, v) and (v, w)
for some u, v, w E V. Our goal will be to construct e' so that it maps into the set of
linear extensions of the almost bipartite orientation Ov,, in which our directed-path
(u, v), (v, w) is the unique directed-2-path. To find 8', first consider the bipartite
graph G' with vertex-set V\{v} and edge-set E\ ({u, v} U {v, w}) U {u, w}, along
with the orientation 0' of its edges that agrees on common edges with 0 and contains
(u, w). Clearly 0' is acyclic. If f is a linear extension of 0, we regard the restriction
f' of f to V\{v} as a strict order-preserving map on 0', and analogously to the proof
of Theorem 1.2.6, we can transform injectively f' into a strict order-preseving map
g' with Im(g') = Im(f') = Im(f)\{f(v)} of the bipartite orientation of G' that
contains (u, w). Now, if we define g E Bij(V, [n]) via g(x) = g'(x) for all x E V\{v}
and g(v) = f(v), we see that g is a linear extension of Ouv.. We let E'(f) = g.
The technical work for proving the general injectiveness of E', and its non-surjectiveness
when 0 is not almost bipartite, has already been presented in the proof of Theorem 1.2.6: That 0' is injective follows from the injectiveness of the map transforming
f' into g', and then by noticing that f(v) = g(v). Non-surjectiveness follows from
noting that if 0 is not almost bipartite, then 0 contains a directed-2-path (a, b), (b, c)
18
5.
f
3
2
2
5
g
Figure 1-2: An example of the function 0' for the case of odd cycles. Squares show
the numbers that will be flipped at each step. Dashed arrows indicate arrows whose
orientation still needs to be reversed, while dashed-dotted arrows indicate those whose
orientation will never be reversed. In particular, 4 will remain labeling the same vertex
during all steps.
with a, b, c E V and b = v, so we cannot have simultaneously g'(a) = min Im(f') and
g'(c) = max Im (f').
1.3
Comparability graphs.
In this section, we will study our main problem of the chapter using more general
techniques. As a consequence, we will be able to understand the case of comparability
graphs, which includes bipartite graphs as a special case. Let us first recall the main
object of this section:
Definition 1.3.1. A comparability graph is a simple undirected graph G = G(V, E)
for which there exists a partialorder on V under which two different vertices u, v E V
are comparable if and only if {u, v} E E.
The acyclic orientations of the edges of a comparability graph G that maximize the
number of linear extensions are precisely the orientations that induce posets whose
comparability graph agrees with G.
Comparability graphs have been largely discussed in the literature, mainly due to
their connection with partial orders and because they are perfectly orderable graphs
and more generally, perfect graphs. Comparability graphs, perfectly orderable graphs
and perfect graphs are all large hereditary classes of graphs. In Gallai's fundamental work {1967}, a characterization of comparability graphs in terms of forbidden
subgraphs was given and the concept of modular decomposition of a graph was introduced.
19
Note that, given a comparability graph G = G(V, E), we can find at least two
partial orders on V induced by acyclic orientations of E whose comparability graphs
(obtained as discussed above) agree precisely with G, and the number of such posets
depends on the modular structure of G. Let us record this idea in a definition.
Definition 1.3.2. Let G = G(V, E) be a comparabilitygraph, and let 0 be an acyclic
orientation of E such that the comparability graph of the partial order of V induced
by 0 agrees precisely with G. Then, we will say that 0 is a transitive orientation of
G.
We will present two methods for proving our main result. The first one (Subsection 1.3.1) relies on Stanley's transfer map between the order polytope and the chain
polytope of a poset, and the second one (Subsection 1.3.2) is made possible by relating
our problem to network flows.
1.3.1
Geometry.
To begin, let us recall the main definitions and notation related to the first method.
Definition 1.3.3. We will consider R 1"] with euclidean topology, and let {ej}E[,l] be
the standard basis of RH"]. For J C [n], we will define ej := E ,ej and eo := 0;
furthermore, for x c R nI we will let xj := ZjEJ x3 and xO := 0.
Definition 1.3.4. Given a partial order P on [n), the order polytope of P is defined
as:
0 (P) := {x E RI'n : 0 < xi
1 and xj 5
xk
whenever j
p k, V i, j, k E [n]}.
The chain polytope of P is defined as:
C (P) := {x E RI'I : xi > 0, V i E [n] and XC
K
1 whenever C is a chain in P}.
Stanley's transfer map cJ : 0 (P) -+ C (P) is the function given by:
Cfx)
if i is not minimal in P,
xi - maxjoi x3
if i is minimal in P.
xi
Let P be a partial order on [n]. It is relatively simple to see from the definitions
that the vertices of 0 (P) are given by all the ej with I an order filter of P, and those
of C (P) are given by all the eA with A an antichain of P.
Now, a well-known result of Stanley {1986} states that Vol (0 (P)) = Le(P)
where e(P) is the number of linear extensions of P. This result can be proved by
considering the unimodular triangulation of 0 (P) whose maximal (closed) simplices
x 1(n) _< 1} with
X,-1( 2 ) < ...
have the form A, := {x E RIn] : 0 < X,-1(i)
u : P -+ n a linear extension of P. However, the volume of C (P) is not so direct
to compute. To find Vol (C (P)), Stanley made use of the transfer map 4, a pivotal
idea that we now wish to describe in detail since it will provide a geometrical point
of view on our main problem.
20
It is easy to see that 4) is invertible and its inverse can be described by:
4r'(x)i =
max
C chain in P:
i is maximal in C
xC, for all i e [n] and x E C (P).
As a consequence, we see that dr 1 (eA) = eAv for all antichains A of P, where AV
is the order filter of P induced by A. It is also straightforward to notice that 4) is
linear on each of the A, with u a linear extension of P, by staring at the definition
of A,. Hence, for fixed u and for each i E [n], we can consider the order filters
AY := ou-([i, n]) along with their respective minimal elements Ai in P, and notice
that 4)(eAY) = eA, and also that ct(0) = 0. From there, 4) is now easily seen to be
a unimodular linear map on A., and so Vol (4) (A,)) = Vol (A,) = 3. Since 4) is
invertible, without unreasonable effort we have obtained the following central result:
Theorem 1.3.5 (Stanley {1986}). Let P be a partialorder on [n]. Then, Vol(0 (P)) =
Vol(C (P)) = le(P), where e(P) is the number of linear extensions of P.
Definition 1.3.6. Given a simple undirected graph G
=
G([n], E), the stable poly-
tope STAB(G) of G is the full dimensional polytope in R [ obtained as the convex
hull of all the vectors e 1 , where I is a stable (a.k.a. independent) set of G.
Now, the chain polytope of a partial order P on [n] is clearly the same as the
stable polytope STAB (G) of its comparability graph G = G([n], E) since antichains
of P correspond to stable sets of G. In combination with Theorem 1.3.5, this shows
that the number of linear extensions is a comparability invariant, i.e. two posets with
isomorphic comparability graphs have the same number of linear extensions.
We are now ready to present the first proof of the main result for comparability
graphs. We will assume connectedness of G for convenience in the presentation of the
second proof.
Theorem 1.3.7. Let G = G(V, E) be a connected comparability graph. Then, the
acyclic orientations of E that maximize the number of linear extensions are exactly
the transitive orientations of G.
First proof. Without loss of generality, assume that V = [n]. Let 0 be an acyclic
orientation of G inducing a partial order P on [n]. If two vertices i, j E [n] are
incomparable in P, then {i, j} V E. This implies that all antichains of P are stable
sets of G, and so C (P) C STAB (G).
On the other hand, if 0 is not transitive, then there exists two vertices k, f E [n]
such that {k, f} V E, but such that k and t are comparable in P, i.e. the transitive
closure of 0 induces comparability of k and f. Then, ek + et is a vertex of the stable
polytope STAB (G) of G, but since C (P) is a subpolytope of the n-dimensional cube,
ek + ee V C (P). We obtain that C (P) $ STAB (G) if 0 is not transitive, and so
C (P) ; STAB (G).
If 0 is transitive, then C (P) = STAB (G). This completes the proof.
21
1.3.2
Poset theory.
Let us now introduce the background necessary to present our second method. This
will eventually lead to a different proof of Theorem 1.3.7.
Definition 1.3.8. If we consider a simple connected undirected graph G = G(V, E)
and endow it with an acyclic orientation of its edges, we will say that our graph is an
oriented graph and consider it a directed-graph, so that every member of E is regarded
as an ordered pair. We will use the notation Go = Go(V, E) to denote an oriented
graph defined in such a way, coming from a simple graph G.
Definition 1.3.9. Let Go = GO(V, E) be an oriented graph. We will denote by Go
the oriented graph with vertex-set V := V U {, } and set of directed-edges E equal
to the union of E and all edges of the form:
(v,1) with v G V and outdeg (v) = 0 in Go, and
(0, v) with v G V and indeg (v) = 0 in Go.
A natural flow on Go will be a function f : E -+ N such that for all v E V, we have:
Z
(xv)
f(x, v)=
E
f(v, y).
(vy)
E
In other words, a natural flow on Go is a nonnegative integer network flow on Go
with unique source 0, unique sink 1, and infinite edge capacities.
First, let us relate natural flows on oriented graphs with linear extensions of induced posets.
Lemma 1.3.10. Let Go = GO(V, E) be an oriented graph with induced partial order
P on V, and with |VI = n. Then, the function g : E -+ N defined by
= I c: o- is a linear extension of P and a(u) = u(v)
if (u, v) E E,
g(v,1)
=
-
g(u, v)
o is a linear extension of P and o(v)
if v
g(0, v)
=
=
n}I
C V and outdeg (v) = 0 in Go, and
I{
: o- is a linear extension of P and u(v) = 1
if v E V and indeg (v) = 0 in Go,
is a naturalflow on Go. Moreover, the net g-flow from 0 to 1 is equal to e(P).
Proof. Assume without loss of generality that V = [n], and consider the directedgraph K on vertex-set V(K) = [n] U {0, i} whose set E(K) of directed-edges consists
of all:
(i,j) for i <p j,
(i, j) and (j, i) for illpj,
(0, i) for i minimal in P, and
(i, 1
for i maximal in P.
22
As directed-graphs, we check that G, is a subgraph of K. We will define a network
flow on K with unique source 0 and unique sink 1, expressing it as a sum of simpler
network flows.
First, extend each linear extension u of P to V(K) by further defining a (o) = 0
and a (i) = n + 1. Then, let
f,: E(K)
f (X, Y)
=
-+ N be given by
1 if c-(x) = c-(y) -1,
0 otherwise.
Clearly, f0 defines a network flow on K with source 0, sink 1, and total net flow 1,
f, defines a network flow on K with total net flow e(P).
E
and then f :=
a linear ext.
of P
Moreover, for each (x, y) E E we have that f(x, y) = g(x, y). It remains now to check
that the restriction of f to E is still a network flow on G0 with total flow e(P).
We have to verify two conditions. First, for i, j E [n] and if ii pj, then
I{u : a is a lin. ext. of P and u(i) = u(j) - 1}I
-
{u:
-isalin. ext. of P and u(j) = u(i)-1}|,
so f (i, j) = f(j, i), i.e. the net f-flow between i and j is 0. Second, again for i, j E [n],
if i <p j but i 4 pj, then f(i, j) = 0. These two observations imply that g defines a
network flow on Go with total flow e(P).
The next result was obtained in Edelman et al. {1989} using the theory of promotion
and evacuation for posets, and their proof bears no resemblance to ours.
Corollary 1.3.11. Let P be a partial order on V, with IVI = n. If A is an antichain
of P, then e(P) _ EvEAe(P\v), where P\v denotes the induced poset on V\{v}.
EvEs e(P\v). Moreover, if I is a subset
Similarly, if S is a cutset of P, then e(P)
of V that is either a cutset or an antichain of P, then e(P) = ZvEI e(P\v) if and
only if I is both a cutset and an antichain of P.
Proof. Let G = G(V, E) be any graph that contains as a subgraph the Hasse diagram
of P, and orient the edges of G so that it induces exactly P to obtain an oriented
graph Go. Let g be as in Lemma 1.3.10. Since edges representing cover relations of
P are in G and are oriented accordingly in Go, the net g-flow is e(P). Moreover, by
the standard chain decomposition of network flows of Ford Jr and Fulkerson {2010}
(essentially Stanley's transfer map), which expresses g as a sum of positive flows
through each maximal directed-path of Go, it is clear that for A an antichain of
P, we have that e(P) > EvCA Z(xv)E g(x, v), since antichains intersect maximal
directed-paths of G at most once. Similarly, for S a cutset of P, we have that
e(P)
Eves E(xv)EZ g(x, v) since every maximal directed-path of G0 intersects S.
Furthermore, equality will only hold in either case if the other case holds as well.
But then, for each v E V, the map Trans that transforms linear extensions of P\v
23
into linear extensions of P and defined via: For u- a linear extension of P\v and
K := max a(y),
y <p V
K+ 1
Trans (u) (x) =
0(x)
ao(x)
+ 1
if x = v,
if U(X) >
K,
otherwise,
is a bijection onto its image, and the number E(,v)cE g(x, v) is precisely Im (Trans) 1.
E
Getting ready for the second proof of Theorem 1.3.7, it will be useful to have a
notation for the main statistic of study in this chapter:
Definition 1.3.12. Let G = G(V, E) be an undirected simple graph. The maximal
number of linear extensions of a partial order on V induced by an acyclic orientation
of E will be denoted by F(G).
Second proof of Theorem 1.3.7. Assume without loss of generality that V = [n].
We will do induction on n. The case n = 1 is immediate, so assume the result holds
for n - 1. Note that every induced subgraph of G is also a comparability graph and
moreover, every transitive orientation of G induces a transitive orientation on the
edges of every induced graph of G. Now, let 0 be a non-transitive orientation of
E with induced poset P, so that there exists a comparable pair {k, f} in P that is
stable in G. Let S be an antichain cutset of P. Then, S is a stable set of G. Letting
G\i be the induced subgraph of G on vertex-set [n]\{i}, we obtain that e(G) >
EiEs e(P\i) = e(P), where the first inequality is an application of
iES (G\i)
Corollary 1.3.11 on a transitive orientation of G, along with Definition 1.3.12 and
the inductive hypothesis, the second inequality is obtained after recognizing that the
poset induced by 0 on each G\i is a subposet of P\i and by Definition 1.3.12, and
the last equality follows because S is a cutset of P. If SI > 1 or S n {k, f} = 0, then
by induction the second inequality will be strict. On the other hand, if S = {k} or
S = {}, then the first inequality will be strict since {k, t} is stable in G.
Lastly, the different posets arising from transitive orientations of G have in common that their antichains are exactly the stable sets of G, and their cutsets are exactly
the sets that meet every maximal clique of G at least once, so by the corollary, the
inductive hypothesis and our choice of S above, these posets have the same number
of linear extensions and this number is in general at least EiES e(G\i), and strictly
greater if S
{k} or S ={.
1.4
Beyond comparability and enumerative results.
In this section, we will firstly illustrate a short application of the ideas developed
in Section 1.3 to the case of odd cycles, re-establishing Theorem 1.2.8 using a more
elegant technique (Subsection 1.4.1) that applies to other families of graphs. Then, in
24
,
Subsection 1.4.2, we will obtain basic enumerative results for e(G). Finally, in Subsection 1.4.3, we will study the random variable e(G) when G is a random graph with
distribution G,,, 0 < p < 1. As it will be seen, if G ~ G,,, then log 2 E(G) concentrates tightly around its mean, and this mean is asymptotically equal to n log 2 logb n 2
where b = 1.
This will allow us to obtain, for the case of random graphs, new bounds
for the volumes of stable polytopes, and a very strong concentration result for the
entropy of a graph, both of which will hold a.s..
1.4.1
A useful technique.
We start with two simple observations that remained from the theory of Section 1.3.
Firstly, note that for a general graph G, finding e(G) is equivalent to finding the
chain polytope of maximal volume contained in STAB (G), hence:
Observation 1.4.1. For a simple graph G, we have:
e(G)
n! Vol(STAB(G)).
Also, directly from Theorem 1.3.7 we can say the following:
Observation 1.4.2. Let P and Q be partial orders on the same ground set, and
suppose that the comparability graph of P contains as a subgraph the comparability
graph of Q. Then, e(Q) ;> e(P) and moreover, if the containment of graphs is proper,
then e(Q) > e(P).
Second proof of Theorem 1.2.8. Note that every acyclic orientation 0 of E induces a partial order on V whose comparability graph contains (as a subgraph) the
comparability graph of a poset given by an almost bipartite orientation, and this
containment is proper if 0 is not almost bipartite. By the symmetry of G, then all
of the almost bipartite orientations are equivalent.
Note to proof: The same technique allows us to obtain results for other restrictive
families of graphs, like odd cycles with isomorphic trees similarly attached to every
element of the cycle or, perhaps more importantly, odd anti-cycles. These results,
however, are not included in this thesis.
1.4.2
General bounds for the main statistic.
Let us now turn our attention to the general enumeration problem. Firstly, we need to
dwell on the case of comparability graphs, from where we will jump easily to general
graphs.
Theorem 1.4.3. Let G = G(V, E) be a comparability graph, and further let V =
{v 1, v2 ,... ,vn}. For u 1 ,u 2 ,... , uk E V, let G\ulu2 ... uk be the induced subgraph of
G on vertex-set V\{ui, u2 ,... , uk}. Then,
-(G)
X(G)X(G\v1)x(G\v
25
iV,2)x(G\veiV,2V3)
... X(Vcn)'
where 6, denotes the symmetric group on [n] and x(.) denotes the chromatic number
of the graph.
Proof. Let us first fix a perfect order w of the vertices of G, i.g. w can be a linear
extension of a partial order on V whose comparability graph is G. Let H be an
induced subgraph of G with vertex-set V(H) and edge-set E(H), let WH be the
restriction of w to V(H), and let Q be the partial order of V(H) given by labeling
every v EV(H) with WH(v) and orienting E(H) accordingly. Using the colors of the
optimal coloring of H given by WH, we can find X(H) mutually disjoint antichains of
Q that cover Q, so by Corollary 1.3.11 we obtain that
e(Q)
(
e(Q\v).
(1.4.1)
X(H)H
Now, we note that each Q\v with v E V(H) is also induced by the respective restriction of w to V(H)\v, and that the comparability of Q\v is H\v, and then each of
the terms on the right hand side can be expanded similarly. Starting from H = G
above and noting the fact that E(G) = e(Q) for this case, we can expand the terms
of 1.4.1 exhaustively to obtain the desired expression.
Corollary 1.4.4. Let G = G(V, E) be any graph on n vertices with chromatic number
k := X(G). Then
(G)
knk!'
Proof. We can follow the proof of Theorem 1.4.3. This time, starting from H = G, Q
will be a poset on V given by a minimal coloring of G, i.e. we color G using a minimal
number of totally ordered colors and orient E accordingly. Then, F(G)
e(Q) and we
can expand the right hand side of 1.4.1, but noting that Q\v can only be guaranteed
to be partitioned into at most X(G) antichains, and that the chromatic number of a
graph is at most the number of vertices of that graph.
L1
Noting that the number of cutsets is a least 2 in most cases, a similar argument
to that of Theorem 1.4.3 implies:
Observation 1.4.5. Let G = G(V, E) be a connected graph. Then:
F_(G) < 2 1:F(G\v).
vEV
.
Example 1.4.6. If G = G(V, E) is the odd cycle on 2n + 1 vertices, then for each
v E V we have E(G\v) = E2n, the (2n)-th Euler number, and X(G) = 3, so a :=
(2n + 1)E2 n >
(2n + 1)!
an
4 (6 )2n
b.
32-2 . 3!. As n goes to infinity, then
2
t(G)
Other upper bounds can be obtained from rather different considerations.
26
Proposition 1.4.7. Let G = G(V, E) be a simple graph on n vertices. Then, F(G) is
at most equal to the number of acyclic orientations of the edges of C, the complement
of G. Equality is attained if and only if G is a complete p-partite graph, p E [n].
Proof. Let E be the set of edges of C, so that E Li E = ().
The inequality holds since two different linear extensions (understood as labelings
of V with the totally ordered set [n]) of the same acyclic orientation of E induce
different acyclic orientations of (V) = E Li E: As both induce the same orientation of
E, they must induce different orientations of E.
To prove the equality statement, first note that if G is not a complete p-partite
graph, then there exist edges {a, b}, {a, c} E E such that {b, c} E E. Suppose that
(b, c) is a directed-edge in an optimal orientation 0 of E. Then, if we label the vertices
of Z with the (totally ordered) set [n] in such a way that c < a < b comparing vertices
according to their labels, our labeling induces an acyclic orientation of E which cannot
be obtained from a linear extension of 0. Hence, 6(G) is strictly less than the number
of acyclic orientations of E.
If G is a complete p-partite graph, then suppose that there exists an acyclic
orientation 0 of E that cannot be obtained from a linear extension of 0, where 0 is
any optimal orientation of E. Then, in the union of the (directed-)edges in both 0
and 0, we can find a directed-cycle that uses at least one (directed-)edge from both
o and 0. Take one such directed-cycle with minimal number of (directed-) edges.
As G is a comparability graph, then 0 is transitive, and so the directed-cycle has the
form E1 P1 E2 P2 ... EmPm, where Ej is a directed-edge in 0, P is a directed-path in
0, and m > 1. Let E1 = (a, b), and let (b, c) be the first directed-edge in P along
the directed-cycle. Since G is complete p-partite, then {a, c} E E because {b, c} E E.
Since 0 is transitive, (a, c) must be a directed-edge in 0. However, this contradicts
the minimality of the directed-cycle.
1.4.3
Random graphs.
.
Changing the scope towards probabilistic models of graphs, specifically to G ,,, we
will obtain a tight concentration result for the random variable e(G) with G .' G
The central idea of the argument will be to choose an acyclic orientation of a graph
G - Gn,, from a minimal proper coloring of its vertices. We expect this orientation
to be nearly optimal.
Let us first recall two remarkable results that will be essential in our proof. The
first one is a well-known result of Bollobhs, later improved on by McDiarmid:
Theorem 1.4.8 (Bollobis {1988}, McDiarmid {1990}). Let G ~ G, with 0 < p < 1,
=
1
.
Then:
x(G)
= 2 log n -
2
n
as
logb logon + 0(1)
'
and define b
where X(G) is the chromatic number of G.
27
To state the second result, we first need to introduce the concept of entropy of a
convex corner, originally defined in CsiszAr et al. {1990}. We only present here the
statement for the case of stable polytopes of graphs.
Definition 1.4.9. Let G = G([n, E) be a simple graph, and let STAB (B) be the
stable polytope of G. Then, the entropy H(G) of G is the quantity:
H(G):=
min
n
aESTAB(G)
log 2 a.
In 1995, Kahn and Kim proved certain bounds for the volumes of convex corners
in terms of their entropies. One of them, when applied to stable polytopes, reads as
follows:
Theorem 1.4.10 (Kahn and Kim {1995}). Let G = G([n], E) be a simple graph, and
let STAB(G) be the stable polytope of G. Then:
nn2nH(G) > n!Vol(STAB(G)) > n!2-HG
Equipped now with these background results, the following is true:
-
G,, with 0 < p < 1, b =- 1
, and write s = 2logbn
-
Theorem 1.4.11. Let G
20log logbn. Then:
log 2 e(G)
Also, E[ log 2 e(G)]
n log 2 s holds a.s..
nlog 2 s.
-
Proof. Let n tend to infinity. Consider the chromatic number of the graph G Gn,
and color G properly using k = X(G) colors, say with color partition a1 +a 2 +- - -+ak =
--log2 ak!
> k0log2 [2J!. By Theorem
n. Then
1.4.8, log
we know
2 E(G)
that k
s
n
n
+ 0(1)
log 2 a 1 !+
+
s so:
, so
n log 2 s -
log 2 e(G)
2
1n 2
+ n(log 2 s)
2s
+0
( )
s.
a.s..
(1.4.2)
We remark here that inequality 1.4.2 gives a slightly better bound than the one
obtained directly from Corollary 1.4.4.
Now, the function log 2 - satisfies the edge Lipschitz condition in the edge exposure
martingale since addition of a single edge to G can alter E by a factor of at most 2,
so we can apply Azuma's inequality to obtain:
Pr[ 1log 2
E(G)
2
- E[ log 2 F(G)]j > n (log2 logbn)" ] < logn
Combining these two results, we see that:
E[ log 2
(G)]
(nlog 2 s)(1 + o(1)),
28
H(G)
in (- log 2 a )
=
-
log2
n/ ai)
\
-
log 2 !oc(G)
n
=
log 2
(
and moreover, that log 2 (G) ~ E[ log 2 e(G)] a.s. holds.
The second necessary inequality comes, firstly, from using Observation 1.4.1, so
that e(G) < n!Vol (STAB (G)), and then from a direct application of Theorem 1.4.10.
log 2 e(G). Now, we further observe that for
We obtain that n(log 2 n - H(G))
1- y(G), and then:
a E STAB (G), we have Ej -a.
s + c holds
A classic result of Grimmett and McDiarmid {1975} states that X(G)
a.s., where c =2 log b+1. Hence, a.s., H(G)
(log 2
g)=
log 2 n-log 2 (s+0(1)),
and then nlog 2 (s + 0(1)) : log 2 E(G). From here, we directly obtain:
log 2
F(G)
nlog2 s+O
(n)
(1.4.3)
a.s..
Therefore, from inequalities 1.4.2 and 1.4.3:
log 2 F-(G) = nlog2 s+O(n) a.s..
Calculating inequality 1.4.3 more precisely by dropping the 0-notation and using
Grimmett and McDiarmid's constant, we obtain:
n!
e
Sns"
2/(0og2 b)
n Vol(STAB(G)) < - . Cnj* a.s., where c = 2
n!2
Corollary 1.4.13. Let G ~ GnP with 0 < p < 1, b = n
2 logo logo n. Then, for large enough n:
log 2
1.5
+
+
H(G) 5 log 2 (
+
and s = 2log n
-
(1n
logn
.
Sn
2
-
Corollary 1.4.12. Let G - G,, with 0 < p < 1, b = 11P and s =
2 logo logo n. Then, for large enough n:
a.s..
Further techniques.
In this section, we will see how the main problem of the chapter has two more presentations as selecting a region in the graphical arrangement with maximal fractional
volume, or as selecting a vertex of the graphical zonotope that is farthest from the
origin in Euclidean distance.
29
Definition 1.5.1. Consider a simple undirected graph G = G([n], E). The graphical
arrangement of G is the central hyperplane arrangement in IRH1 given by:
AG = Ix
C Rn] : xi - xj = 0 , for some {i, j}
E}.
The regions (see Definition 2.2.1 and the comments thereafter) of the graphical
arrangement AG with G = G([n], E) are in one-to-one correspondence with the acyclic
orientations of G (Proposition 2.2.2). Moreover, the complete fan in R[n] given by AG
is combinatorially dual to the graphicalzonotope of G:
Za"'''
=
[e
-
e3 , e
-
e1,
{i,j}EE
and there is a clear correspondence between the regions of AG and the vertices of
Zntral.
Following Klivans and Swartz {2011}, we define the fractional volume of a region
R of AG to be:
Vol' (R)
-
Vol (B n nR)
Vol (Bn)
where B" is the unit n-dimensional ball in R [n.
With little work, using Proposition 2.2.2 and the symmetry of the permutohedron,
it is possible to say the following about these volumes:
Proposition 1.5.2. Let G = G([n], E) be an undirected simple graph, and let AG be
its graphical arrangement. If R is a region of AG and P is its corresponding partial
order on [n] under the map of Proposition 2.2.2, then:
e(P)
n!
Volf (R)= n!
The problem of finding the regions of AG with maximal fractional volume is, intuitively, closely related to the problem of finding the vertices of Z* t ral that are farthest
from the origin under some appropriate choice of metric. It turns out that, with Euclidean metric, a precise statement can be formulated when G is a comparability
graph:
Theorem 1.5.3. Let G = G(V, E) be a comparability graph. Then, the vertices of
the graphical zonotope of Zcntral that have maximal Euclidean distance to the origin
are precisely those that correspond to the transitive orientations of E, which in turn
have maximal number (G) of linear extensions.
To prove Theorem 1.5.3, we first note that for a simple (undirected) graph G =
G(V, E), the vertex of Zn t a corresponding to a given acyclic orientation of E is
precisely the point:
(outdeg (v) - indeg(v))vEV,
where outdeg (.) and indeg (.) are calculated using the given orientation.
We need to establish a preliminary lemma.
30
Lemma 1.5.4. Let Go = Go(V, E) be an oriented graph. Then,
(indeg (v) - outdeg (v)) 2 = E + tri (Go) + incom (Go) - com (Go),
vEV
where:
1. tri (Go) is the number of directed-triangles(u, v), (v, w), (u, w) E E.
2. incom(Go) is the number of triples u,v,w C V such that (v,w), (w,v) 0 E but
either (u, v), (u, w) E E or (v, u), (w, u) E E.
E E such that (u, w)
$
3. com (Go) is the number of directed-2-paths (u, v), (v, w)
E.
Proof. For v E V, outdeg (v) 2 is equal to outdeg (v) plus two times the number of
pairs u / w such that (v, u), (v, w) E E, indeg (v) 2 is equal to indeg (v) plus two times
the number of pairs u, 74 w such that (u, v), (w, v) E E, and outdeg (v) - indeg (v) is
equal to the number of pairs u 74 w such that (u, v), (v, w) E E. If we add up these
terms and cancel out terms in the case of directed-triangles, we obtain the desired
equality.
An important consequence of Lemma 1.5.4 is the following:
If G = G(V, E) is a simple graph, all the
in their values of tri(-) and of |E|, which
incom (.). Moreover, com (-) + incom (-)
G of the form {u, v}, {v, w} E E with u
choice of orientation for E.
acyclic orientations of E will not vary
depend on G, but only in com (.) and
is equal to the number of 2-paths in
74 w, so it is also independent of the
Proof of Theorem 1.5.3. We apply Lemma 1.5.4 directly. Since G is a comparability graph, from Theorem 1.3.7, we know that the value of incom (.) - com (-)
will be maximized precisely on the transitive orientations of G, since all transitive
orientations force com (-) = 0.
Remark 1.5.5 (To Theorem 1.5.3). In fact, per the Second proof of Theorem 1.2.8
in Subsection 1.4.1, the analogous result to Theorem 1.5.3 also holds true for odd
cycles. In joint work with Kyle Gettig, we have again made use of Observation 1.4.2
to prove that odd anti-cycles (i.e. the complement graphs to odd cycles) also have
this property.
31
32
Chapter 2
Largest Eigenvalue of the Laplacian
Matrix.
2.1
Introduction.
Let G = G([n], E) be a simple (undirected) graph, where [n] = {1, 2,..., n}, n
The adjacency matrix of G is the n x n matrix A = A(G) such that:
(
=
1
C P.
if {i,j} E E,
A
The Laplacian matrix of G is the n x n matrix L = L(G) such that:
(L)=
di
if i = j,
:
where di = dG(i)= (dG)i is the degree of vertex i in G.
The spectral theory of these matrices, i.e. the theory about their eigenvalues and
eigenspaces, has been an object of much study for the last 40 years. The roots of
this beautiful theory, however, can arguably be traced back to Kirchhoff's matrixtree theorem, whose first proof is often attributed to Borchardt {1860} even though
at least one proof was already known by Sylvester {1857}. A recollection of some
of the interesting applications of the theory can be found in Spielman {2009}, and
more complete accounts of the mathematical backbone are Brouwer and Haemers
{2011} and Chung {1997}. Still, it would be largely inconvenient and prone to unfair
omissions to attempt here a brief account of the many past and present contributors
and contributing papers to the modern spectral theory of graphs, and we refer the
reader to our references for further inquiries of the literature.
This chapter aims to fill one (of the many) gap (s) in our present knowledge of the
spectrum of the Laplacian matrix, namely, the lack of results about its eigenvectors
with largest eigenvalue. We will answer the question: What information about the
structure of a graph is carried in these eigenvectors? This question will be studied
while remaining loyal to the central theme of this thesis: The mathematics of acyclic
33
axAmax(G),
be the (real) eigenvalues of L, and note that A 2 > 0 if G is a connected graph; we have
effectively dropped G from the notation for convenience but remark that eigenvalues
and eigenvectors depend on the particular graph in question, which will be clear from
the context. We will also let EA, be the eigenspace corresponding to A. In its most
primitive form, Fiedler's nodal domain's theorem {Fiedler, 1975} states that when
G is connected and for all x e EA 2 , the induced subgraph G [{i E [n] : xi > 0}] is
connected. Related ideas and results might be found in Merris {1998}.
On this chapter, we will go even further in the way in which eigenvectors of the
Laplacian may be used to learn properties of G. To explain this, let us firstly call a
map,
0 :E -+ ([n] x [n]) U E
=
[n] 2 U E,
such that O(e) c {e, (i, j), (j, i)} for all e := {i, j} E E, an (partial) orientation
of E (or G), and say that, furthermore, 0 is acyclic if O(e) $ e for all e and the
directed-graph on vertex-set [n] and edge-set O[E] has no directed-cycles. Along the
exposition of the chapter, we will oftentimes also identify the object 0 with the set
0[E].
During this second part of the thesis, eigenvectors of the Laplacian and more precisely, elements of Ekmax, will be used to obtain orientations of certain (not necessarily
induced) subgraphs of G. Henceforth, given G and for all x c R[n, the reader should
always automatically consider the orientation (map) Ox = Ox(G) associated to x,
Ox: E -+ [n] 2 U E, such that for e := {ij} c E:
if Xi = X3
e
,
- -"n =
orientations of graphs. Our work follows the spirit of Fiedler {2011}, who pioneered
the use of eigenvectors of the Laplacian matrix to learn about a graph's structure.
One of the first observations that can be made about L is that it is positivesemidefinite, a consequence of it being a product of incidence matrices. We will thus
let,
0 = A
A2 <
.
Ox(e) =
(i, j) if xi < xj,
(j, i)
if Xi > Xj.
The orientation Ox will be said to be induced by x (e.g. Figure 2-1c).
Implicit above is another subtle perspective that we will adopt, explicitly, that
vectors x E RMn] are real functions from the vertex-set of the graph in question (all
our graphs will be on vertex-set [n]). In our case, this graph is G, and even though
accustomed to do so otherwise, entries of x should be really thought of as being
indexed by vertices of G and not simply by positive integers. Later on in Section 2.3,
for example, we will regularly state (combinatorial) results about the fibers of x when
x belongs to a certain subset of RM"I (e.g. Emax), thereby regarding these fibers as
vertex-subsets of the particular graph being discussed at that moment.
Using this perspective, we will learn that the eigenspace ENx. is closely related
to the theory of modular decomposition of Gallai {1967}; orientations induced by
elements of EXax lead naturally to the discovery of modules. This connection will
34
most concretely be exemplified when G is a comparability graph, in which case these
orientations iteratively correspond to and exhaust the transitive orientationsof G. It
will be instructive to see Figure 2-1 at this point.
In Section 2.2, we will introduce the background and definitions necessary to
state the precise main contributions of this chapter. These punch line results will
then be presented in Section 2.3. The central theme of Section 2.3 will be a stepwise
proof of Theorem 2.3.1, our main result for comparability graphs, which summarily
states that when G is a comparability graph, elements of Ex... induce transitive
orientations of the copartition subgraph of G. It will be along the natural course
of this proof that we present our three main results that apply to arbitrary simple
graphs: Propositions 2.3.10 and 2.3.11, and Corollary 2.3.12.
Finally, in Section 2.4, we will present a curious novel characterization of comparability graphs that results from the theory of Section 2.3.
2.2
Background and definitions.
Let us first review the mathematical background relevant to this chapter. A few of
the concepts that we will present here have already been introduced in Chapter 1,
but we will restate them to free the reader from having to skip repeatedly over a
large number of pages. In any case, we will also add small further clarifications to the
original setting of Chapter 1, adapting the narrative of this chapter to fit our current
needs.
2.2.1
The graphical arrangement.
Definition 2.2.1. Let G = G([n], E) be a simple (undirected) graph. The graphical
arrangement of G is the union of hyperplanes in RlR:
AG := {x G RI'l : xi - xj = 0 , for some {i,j} E E}.
Basic properties of graphical arrangements and, more generally, of hyperplane
arrangements, are presented in Chapter 2 of Stanley {2004}.
For G as in Definition 2.2.1, let R(AG) be the collection of all (open) connected
components of the set RHl\Ac. An element of R(AG) is called a region of AG, and every region of AG is therefore an n-dimensional open convex cone in RI"1. Furthermore,
the following is true about regions of the graphical arrangement:
Proposition 2.2.2. Let G be as in Definition 2.2.1. Then, for all R
x, y E R, we have that:
C R((AG)
and
OR:= OX - O.
Moreover, the map R '-+ OR from the set of regions of AG to the set of orientations
of E is a bijection between R(AG) and the set of acyclic orientations of G.
Motivated by Proposition 2.2.2 and the comments before, we will introduce special
notation for certain subsets of R "I obtained from AG-
35
Notation 2.2.3. Let G be as in Definition 2.2.1. For an acyclic orientation0 of E,
we will let CO denote the n-dimensional closed convex cone in R 'n that is equal to the
topological closure of the region of AG corresponding to 0 in Proposition2.2.2.
2.2.2
Modular decomposition.
We need to concur on some standard terminology and notation from graph theory,
so let G = G([n], E) be a simple (undirected) graph and X a subset of [n].
As customary, C denotes the complement graph of G. The notation N(X) denotes
the open neighborhood of X in G:
N(X)
{j E [n]\X: there exists some i E X such that {i,j} E E}.
The induced subgraph of G on X is denoted by G[X], and the binary operation of
graph disjoint union is represented by the plus sign +. Lastly, for Y C [n], X and Y
are said to be completely adjacent in G if:
X n Y = 0, and
for all i E X and
j
E Y, we have that {i, j} E E.
The concepts of module and modular decomposition in graph theory were introduced by Gallai {1967} as a means to understand the structure of comparability
graphs. The same work would eventually present a remarkable characterization of
these graphs in terms of forbidden subgraphs. Section 2.3 of the present work will
present an alternate and surprising route to modules.
Definition 2.2.4. Let G = G([n], E) be a simple (undirected) graph. A module of G
is a set A C [n] such that for all i, j G A:
N(i)\A = N(j)\A
=
N(A).
Furthermore, A is said to be proper if A C [n], non-trivial if
if G[A] is connected.
Al > 1, and connected
Corollary 2.2.5. In Definition 2.2.4, two disjoint modules of G are either completely
adjacent or no edges exist between them.
Let us now present some basic results about modules that we will need.
Lemma 2.2.6 (Gallai {1967}). Let G = ([n], E) be a connected graph such that Z is
connected. If A and B are maximal (by inclusion) proper modules of G with A
then A n B = 0.
$
B,
Corollary 2.2.7 (Gallai {1967}). Let G = ([n], E) be a connected graph such that C
is connected. Then, there exists a unique partitionof [n] into maximal propermodules
of G, and this partition contains more than two blocks.
36
From Corollary 2.2.7, it is therefore natural to consider the partition of the vertexset of a graph into its maximal modules; the appropriate framework for doing this is
presented in Definition 2.2.8. Hereafter, however, we will assume that our graphs are
connected unless otherwise stated since (1) the results for disconnected graphs will
follow immediately from the results for connected graphs, and (2) this will allow us
to focus on the interesting parts of the theory.
Definition 2.2.8 (Ramirez-Alfonsin and Reed {2001}). Let G = G([n], E) be a
connected graph.
such that:
We will let the canonical partition of G be the set P
=
P(G)
a. If G is connected, P is the unique partition of [n] into the maximal proper
modules of G.
b. If C is disconnected, P is the partitionof [n] into the vertex-sets of the connected
components of C.
Hence, in Definition 2.2.8, every element of the canonical partition is a module of
the graph. Elements of the canonical partition of a graph on vertex-set [8] are shown
in Figure 2-1b.
Definition 2.2.9. In Definition 2.2.8, we will let the copartition subgraph of G be
the graph GP on vertex-set [n] and edge-set equal to:
E\ {{i, j} E E: i, j E A for some A E P}.
2.2.3
Comparability graphs.
We had anticipated the importance of comparability graphs for the results of this
chapter, so let us now recall what they are.
Definition 2.2.10. A comparability graph is a simple (undirected) graph G = G(V, E)
such that there exists a partial order on V under which two different vertices u, v E V
are comparable if and only if {u, v} G E.
A comparability graph on vertex-set [8] = {1, 2, . . , 8} is shown in Figure 2-1b.
Comparability graphs are perfectly orderable graphs and more generally, perfect
graphs. These three families of graphs are all large hereditary classes of graphs.
Note that, given a comparability graph G = G(V, E), we can find at least two
partial orders on V whose comparability graphs (obtained as discussed in Definition 2.2.10) agree precisely with G, and the number of such partial orders depends
on the modular decomposition of G. Let us record this idea in a definition.
Definition 2.2.11. Let G = G(V, E) be a comparabilitygraph, and let 0 be an acyclic
orientation of E. Consider the partial order induced by 0 under which, for u, v E V,
u is less than v iff there is a directed-path in 0 that begins in u and ends in v. If
the comparability graph of this partial order on V (obtained as in Definition 2.2.10)
agrees precisely with G, then we will say that 0 is a transitive orientation of G.
37
(c)
(b)
(a)
1
2
3
4
5
6
1
2
a
-0.1515...
b = -0.2587...
a
c =-0.1021..
d = -0.1866...
d
e0=.8855...
2
8
-a
Figure 2-1: (2-1a) Hasse diagram of a poset P on ground set [8]. (2-1b) Comparability
graph G = G([8], E) of the poset P, where the closed regions depict the maximal
proper modules of G. (2-1c) Unit eigenvector x E Exmax of G fully calculated, where
dim (EAniax) = 1. Arrows represent the induced orientation Ox of G. Notice the
relation between Ox, the modules of G, and the poset P.
2.2.4
Linear algebra.
Some standard terminology of linear algebra and other related conventions that we
adopt are presented here. Firstly, we will always be working in Euclidean space RI'J,
and all (Euclidean-normed real) vector spaces considered are assumed to live therein.
Euclidean norm is denoted by 11 -I1. The standard basis of R[n] will be {ei}[], as
customary. Generalizing this notation, for all I C [n], we will also let:
Zj:=Eei.
iEI
The orthogonal complement in R["I to spanR (e[n]) will be of importance to us, so we
will use special notation to denote it:
R*[n] := (spanR
e[l]))-
For an arbitrary vector space V and a linear transformation T : V -+ V, we will
say that a set U C V is invariant under T, or that T is U-invariant, if T(U) C U.
Lastly, a key concept of this chapter:
C[n],
For a vector x E Rc
and a set
C
we will say that E is a fiber of x
if there exists oc E R such that xi = oc if and only if i E
4.
The notion of being a generic vector in a certain vector space, to be understood
from the point of view of Lebesgue measure theory, is a central ingredient in many of
our results. We now make this notion precise.
Definition 2.2.12. Let V be a linear subspace of R[n] with dim (V) > 0. We will say
that a vector x E V is a uniformly chosen at random unit vector or u.c.u.v.
38
if x is
uniformly chosen at random from the set {y E V : IIyI = 1}.
For x E V a u.c.u.v., a certain event or statement about x is said to occur or hold
true almost surely if it is true with probability one.
2.2.5
Spectral theory of the Laplacian.
We will need only a few background results on the spectral theory of the Laplacian
matrix of a graph. We present these below in a single statement, but refer the reader
to Brouwer and Haemers {2011} for additional background and history.
Lemma 2.2.13. Let G = G([n], E) be a simple (undirected) graph. Let L = L(G)
- <.. An = Amax = Amax(G) be the
be the Laplacian matrix of G and 0 = A, !5 A2
eigenvalues of L. Then:
1. The number of connected components of G is equal to the multiplicity of the
eigenvalue 0 in L.
2. If Z is the complement of G and L is the Laplacian matrix of C, then L =
nI - J - L, where I is the n x n identity matrix and J is the n x n matrix of
all-l's. Consequently, Ama, < n.
3. If H is a (not necessarily induced) subgraph of G on the same vertex-set [n], and
i7n are the eigenvalues of the Laplacian of H, then Ai > iti
...
iL2
if
for all i E [n].
w
Lemma 2.2.13 Part l's proof was discussed during the Introduction (Section 2.1),
and Part 2 is a straightforward verification, but Part 3 is a more advanced result.
2.3
Largest Eigenvalue of a Comparability Graph.
The main goal of this section is to prove the following theorem:
Theorem 2.3.1. Let G = G([n], E) be a connected comparability graph with Laplacian matrix L = L(G) and canonical partition P = P(G). Let Arna = Amax(G) be the
largest eigenvalue of L and EAax. its associated eigenspace. Then, the following are
true:
i. If 0 is a transitive orientation of G, then:
dim (CO n EAmax)
ii. Ex_
iii. Let x
G
1O Co,
= dim (EAmax).
where the union is over all transitive orientations of G.
EAmax. be a u.c.u.v.. Almost surely:
1. If A E -P, then A belongs to a fiber of x.
39
2. If A, A' C P are completely adjacent in G, then A and A' belong to different
fibers of x.
3. x induces a transitive orientation of G'. In particular, G' is a comparability graph.
4. All transitive orientations of G' can be induced by x with positive proba-
bility.
5. If
is a fiber of x, then:
G[ ] = G[B1 ] +
+ G[B,],
where for all i E [k], Bi is a connected module of G and G[Bi] is a comparability graph.
6. G has exactly two transitive orientations if and only if dim (EAx) = 1
and every fiber of x is an independent set of G, so G = GP.
iv. If ? is connected, then dim (EA,.) = 1. If ? is disconnected, then dim (EA,.x)
is equal to the number of connected components of ? minus one.
Remark 2.3.2 (to Theorem 2.3.1). In fact, as it will be explained, all transitive
orientations of G can be obtained with the following procedure: Select an arbitrary
transitive orientation for GP, and select arbitrary transitive orientations for (the
connected components of) each G[A], A E P . Therefore, i-iii imply an iterative
algorithm that obtains every transitive orientation of G with positive probability.
The proof of Theorem 2.3.1 will be stepwise and its notation and conventions will
carry over to the next results, unless otherwise stated. Let us begin with this work.
Proposition 2.3.3. Let G = G([n], E) be a connected comparability graph and let Co
be the (closed convex) cone corresponding to a transitive orientation 0 of G. Then,
Co contains a non-zero eigenvector of L with eigenvalue Aa.. Furthermore:
(Emax)
.
dim (Co n Eax) = dim
Proof. The cases n = 1 and n = 2 are easy to verify, so we assume that n > 2.
The proof consists of two main steps. Firstly, we will prove that Co is invariant under left-multiplication by L. Then, we will prove that dim (Co n Emax)=
dim (EAmax).
Step 1: Lx E Co whenever x
E Co.
Take an arbitrary vector x E Co and let {i,j}
40
c E with (i,3j) in 0.
Hence,
xi < xj. If we consider the vector Lx, then:
(Lx)j
-
(Lx)i = (xdG(j)
Xk) -
-
(x -
Xe)
eEN(i)
kEN(j)
=
E
G(xid(i)
xk) -
(x
1:
-
x)
fEN(i)
kEN(j)
(xj - x)
+
=IN(i) nN(j)I(x - xi)+
EEN(j)\N(i)
5
-
(xi
- xm).
mEN(i)\N(j)
Now, since 0 is transitive and G is comparability, if f E N(j)\N(i), then we must
have that (f, j) is an edge in 0, so that xe 5 x since x E Co. Otherwise, we
would require that {i, } E E, which is false. Similarly, if m E N(i)\N(j), we must
have that (i, m) is an edge in 0, so xm > xi. Since also x3 > xi, then we see that
(Lx)j - (Lx)i > 0. Verification of the analogous condition for every edge of E shows
that indeed Lx E Co.
Step 2: dim (Co n EAmax) = dim
(EAmax)
Suppose on the contrary that dim (Co n Emax) < dim (EAmx). Then, there exists
X*E EAm \spanR (Co n EAmx ) with |Ix*I = 1. Since Co is full-dimensional in R M,
or
we can write x* = x - y for some x, y E C0, where necessarily either x V EL
=
E'
,
then
x*
if
y
E
Otherwise,
In fact, we must have that x, y V E'.
y V E'.
lim LN(x - y)IILN(X _ y)jf = lim LNxIILNxII E Co from Step 1, and similarly,
N-4oo
N-*oo
then x* E -Co, so in both cases x* E spanR (Con Emx.). Hence,
if x E E'
o < |ILNXII, IILNyjj AN max{ ix|I,| |yf|} for all N > 1 and, moreover, since both
LNX IILNxII and LNy/IILNyjj can be made arbitrarily close to spanR (Co n Exmax)
(in particular, using Step 1, each gets close to Co n EAm) for large N, then the same
will be true for
L x
-
LNy
Amax{IxIIyI
= C
max
Therefore, letting N -+ oo, we obtain that x*
our choice of x*, so:
= cx
where c =
1
= 0.
E spanR (Co n EA)ma). This contradicts
EAmax\spanR (Co n Ex.) =0.
E
Lemma 2.3.4. Let G = G([n], E) be a connected comparability graph and let 0 be a
transitive orientation of G. If x E Co n EAmax, x $ 0, satisfies that xu = x, = oc for
some {u, v} E E and oc c R, then there must exist A C [n] such that:
i. A is a (proper non-trivial) connected module of G and u, v c A.
ii. xi = oc for all i
c A.
41
Proof. That such an x may exist is the content of Proposition 2.3.3, but we are
assuming here that indeed, such an x exists with the stated properties.
Consider the maximal (by inclusion) set A C [n] such that G[A] is connected,
u, v c A, and Xk = x for all k E A. Primarily, G[A] cannot be equal to G, since that
would imply that x is equal to te, 1 , which is impossible. Hence, G[A] is a proper
non-trivial connected induced subgraph of G.
We will show that A is a (proper non-trivial connected) module of G. Suppose on
the contrary, that A is not a module of G. Then, there must exist two vertices i, j E A
such that N(i)\A
#
N(j)\A.
Consequently, N(i)AN(j)\A
:
0.
Furthermore,
considering a path in G[A] connecting i and j, we observe that we may assume that i
and j are adjacent in G[A], so that {i, j} E E. Under this assumption, suppose now
that (i, j) is an edge in 0. As 0 is transitive, we must have that (i, k) is an edge in 0
whenever (j, k) is. Similarly, (k, j) must be an edge in 0 whenever (k, i) is. As such,
since N(i)\A
#
N(j)\A, then it must be the case that for k E N(i)AN(j)\A:
If k E N(i), then (i, k) is an edge in 0;
and if k E N(j), then (k, j) is an edge in 0.
Left-Multiplying x by the Laplacian of G, we obtain:
0 = AmaxfX
-
(Lx)j - (Lx)i =
-
z
= AmaxXj
Amax
(
(xj
-xk)
kEN(j)
(xj--
ck)-
kGN(j)\AUN(i)
AmaxXi
-
(xi
-E
-
xt)
eEN(i)
E(xNi-\xA)
tEN(i)\AUN(j)
E
Xi - Xej.
kE N(j)\AUN(i)
Since N(i)AN(j)\A $ 0 and A was chosen maximal, then at least one of the terms
in the last summations must be non-zero and we obtain a contradiction. This proves
that A is a module of G with the required properties.
F1
Theorem 2.3.5. Let G = G([n], E) be a connected comparability graph without proper
non-trivial connected modules. Then:
i. Any x E Exax_\{0} induces a transitive orientation of G.
ii. dim (Ex,x) = 1.
iii. G has exactly two transitive orientations.
Proof. The cases n = 1 and n = 2 are easy to check, so we assume that n > 2.
Fix a transitive orientation 0 of G and consider the cone Co. Per Proposition 2.3.3, we can find at least one x E CO n Exx, x 5 0. By Lemma 2.3.4 and since
42
G does not have proper non-trivial connected modules, x must belong to the interior
of Co. This establishes i.
To prove ii, assume on the contrary, that dim (Ex.x) > 1. Consider two dual
transitive orientations 0 and Odual of G, i.e. Odual is obtained from 0 by reversion
of the orientation of all the edges. Using i, let y, z E EA.x\{0} be such that y E
int(Co), z E int(Codual), and z 0 spanR (y). Then, there exists 0c E (0, 1) such that
0 = cy + (1 - 0C)z E 0 (Co n EA.x), contradicting i.
Finally, iii follows easily from i-ii and Proposition 2.3.3.
The remaining part of the theory will rely heavily on some standard results of the
spectral theory of the Laplacian (Section 2.2.5). These will be of central importance
to establish Proposition 2.3.10, Proposition 2.3.11, and Corollary 2.3.12, which deal
with arbitrary simple graphs.
Lemma 2.3.6. Let G = G([n], E) be a complete p-partite graph with maximal independent sets A,... , Ap. Then, Ana, = n and:
EAx
=
-
{x E R*(n] : If ij
E Aq
spanR (eAq}qe
n R*["].
p])
for some q E [p], then xi = xj}
In particular, dim (EAx.) = p - 1.
Proof. The complement of G has p connected components, so by Parts 1 and 2
in Lemma 2.2.13, Amax = n and dim (Emax.) = p - 1. Let bi,... ,bp E R and let
xE R*[n] be such that xi = bq for all i E Aq, q E [p]. For any i E [n], if i E Aq then
(Lx)i = (n - |Aq|)bq - (0 -|Aq bq) = nbq = nxi. The set of all such x has dimension
p - 1.
Lemma 2.3.7. Let G = G([n], E) be a connected bipartite graph with bipartition
{X, Y}. Then, dim (EAmax) = 1. Furthermore, if x E EA..x\{0}, then either xi < 0
for all i E X and xj > 0 for all j e Y, or vice-versa.
Proof. If G is complete 2-partite, this is a consequence of Lemma 2.3.6. Otherwise,
as a connected bipartite graph, G is also a comparability graph and G does not have
connected proper non-trivial modules, so Theorem 2.3.5 shows that dim (Em.x) = 1
and that x E EAm.\{0} induces a transitive orientation of G. So take x c EAmax\{0}
and suppose that xi = 0, i E X. Then, (Lx)i 74 0 as x induces a transitive orientation
of G and since G is connected.
We have not found an agreed-upon notation in the literature for the following
objects, so we will need to introduce it here.
Definition 2.3.8. Let G = G([n], E) be a simple connected graph, and let Q =
{X 1 ,..., Xm} be a partition of [n] with non-empty blocks. Then, for all k E [im]:
43
a. Gx, will denote the graph on vertex-set [n] and edge-set:
{{i,j}
E: i,j C Xk}.
b. Rxk := {x E R*[] : xi = 0 if i g Xk,i C [n]}.
Also,
RC : = {x E R*ln : x is constant on each Xk, k C [m]}
spanR K e IkE[]) nR*
Observation 2.3.9. In Definition 2.3.8, the linear subspaces RQ and Rx, for all
k G [m], are mutually orthogonal.
Furthermore, any vector x E R*c[n can be uniquely written as:
x = y+x1
with y C RQ and
Xk
+x2 +
-
+xm,
c Rxk, k E [m].
We are now ready to present the results about the space Exax, for simple graphs.
Their proofs will use the same language and main ideas, so we will present them
contiguously to make this resemblance clear.
Proposition 2.3.10. Let G = G([n], E) be a connected simple graph such that C is
connected. For any fixed proper module A of G, the following is true: If x E E ax_,
then A belongs to a fiber of x.
Proposition 2.3.11. Let G = G([n], E) be a connected simple graph such that C is
disconnected. Then, Ama, = n and:
EAmx.
={x E R*1"1 : x, = xj,
whenever i and j belong to the same connected component of ?}.
In particular, dim EAmax is equal to the number of connected components of ? minus one, and G' is a complete p-partite graph, where p is the number of connected
components of G.
Notation for the Proofs of Propositions 2.3.10 and 2.3.11: Let I be the n x
n identity matrix. As usual, P = {A 1 , ... , A,} will be the canonical partition of G.
Let L be the Laplacian matrix of G, LP be the Laplacian matrix of the copartition
subgraph G' of G, and LAq be the Laplacian matrix of GA, for q C [p]. Firstly, we
observe that L = LP + Eq= 1 LAq.
Proof of Proposition 2.3.10. The plan of the proof is to show that the eigenspace
of LP corresponding to its largest eigenvalue lives inside R", and then to show that
this eigenspace is precisely equal to Emax. This will be sufficient since A C Aq for
some q E Ip].
44
To prove the first claim, first note that left-multiplication by L" is Rp-invariant,
where the condition that the Aq's are modules is fundamental to prove this. Now, for
any x E R*[nI, and writing x = y + x, +
-+ x with y E R andxq E RA,, q E [p],
we have that:
L x = Lpy +
S IN(Aq)I
Xq.
q=1
Hence, by Observation 2.3.9, if we can show that the largest eigenvalue of LP is
strictly greater than max{IN(Aq)E}[ej, the claim will follow. This is what we will
do now.
In fact, we will prove that the largest eigenvalue of LP is strictly greater than
max{IN(Aq) I+ IAqI}qE[P]. To check this, first note that both GP and its complement
are connected graphs, and that for q E [p], Aq is both a maximal proper module
and an independent set of GP. For an arbitrary q E [p], consider the (not necessarily induced) subgraph H~q of G on vertex-set Aq U N(Aq) and whose edge-set is
{{i,j} E E : i E Aq and j E N(Aq)}. Firstly, H~q is a complete 2-partite graph, so
its largest eigenvalue is precisely IN(Aq) I + IAqI from Lemma 2.3.6. Secondly, since
both GP and its complement are connected, there exists a (not necessarily induced)
connected bipartite subgraph H of GP such that H~q = H[AqUN(Aq)] and H 5 H~q.
By Lemma 2.2.13 Part 3 and Lemma 2.3.7, the largest eigenvalue of the Laplacian
matrix of H must be strictly greater than that of H~q, since any non-zero eigenvector
for this eigenvalue must be non-zero on the vertices of H that are not vertices of Hq.
Also, by the same Lemma 2.2.13 Part 3, the largest eigenvalue of LP must be at least
equal to the largest eigenvalue of the Laplacian matrix of H. This proves the first
claim.
To prove the second claim, note that for q E [p], left-multiplication by LAq is
RAq-invariant. Also, for an arbitrary x E R*I" decomposed as above, we have that:
Lx = Lpy + E(IN(Aq)I I + LAq)Xq,
q=1
and this gives the unique decomposition of Lx of Observation 2.3.9. But then, from
the proof of the first claim, we note that it suffices to prove that the largest eigenvalue
of LP is strictly greater than that of IN(Aq)i I + LAq for any q E [p]. However, from
Lemma 2.2.13 Part 1, we know that the largest eigenvalue of LA, is at most IAqI, so the
largest eigenvalue of IN (Aq) I1+LAq is at most IN(Aq)+IAq I. We have already proved
that the largest eigenvalue of LP is strictly greater than max{IN(Aq)I + IAqI}qE[p], SO
the second claim follows.
Proof of Proposition 2.3.11. That Gp is a complete p-partite graph is clear, so
from Lemma 2.3.6, it will suffice to prove that EAax is exactly equal to the eigenspace
of LP corresponding to its largest eigenvalue (= n). This is what we do.
As in the proof of Proposition 2.3.10, we observe that left-multiplication by L" is
RP-invariant, and that for q E [p], left-multiplication by LA, is RAq-invariant. For an
45
arbitrary x E R*M with x = y
and noting that IN(Aq)I = n -
+ x, + - -- + x,, where y E R" and Xq E
fAqI
RAq, q E
[Al
in this case, we have that:
p
Lx = Lpy +
Z((n -
|Aq|)I + LAq)Xq,
q=1
and this gives the unique decomposition of Lx of Observation 2.3.9. Hence, we will be
done if we can show that the largest eigenvalue of any of the matrices LAq, q E [p], is
strictly less than JAqI. However, since by construction (from the definition of canonical
partition), G[Aq] satisfies that its complement is connected, then Lemma 2.2.13 Parts
1 and 2 imply that the largest eigenvalue LA, is strictly less than IAqJ , and this holds
for all q E [p]. This completes the proof.
,
Corollary 2.3.12. Let G = G([n], E) be a connected simple graph with canonical
partition P (with L and Ex.. as usual). If LP denotes the Laplacian matrix of G
then the eigenspace of LP correspondingto the largest eigenvalue coincides with EA.x.
Let us now turn back our attention to comparability graphs and to the proofs of
Theorem 2.4.1 and Theorem 2.3.1. Comparability graphs are, as anticipated, specially
amenable to apply the previous two propositions and their corollary. In fact, the
following result already establishes most of Theorem 2.3.1.
Proposition 2.3.13. Let G = G([n], E) be a connected comparability graph with
canonical partition P.
i. For x E EA,x
a u.c.u.v., the following hold true almost surely:
1. If A E P, then A belongs to a fiber of x.
2. If A, A' E P are completely adjacent in G, then A and A' belong to different
fibers of x.
3. x induces a transitive orientation of Gp. In particular, G' is a comparability graph.
4. If
E is a fiber of x, then:
G[E] = G[Bi] +-
+ G[Bkl,
where for all i E k, Bi is a connected module of G and G[Bi] is a comparability graph.
ii. If Z is connected, then dim(Ex) = 1. Also, G' has exactly two transitive
orientations and each can be obtained with probability 1 in i.
iii. If ? is disconnected, then dim (Emax_) = p - 1, where p is the number of connected components of ?. Also, GP has exactly p! transitive orientations and
each can be obtained with positive probability in i.
46
$
Proof. We will work on each case, whether ? is connected or disconnected, separately.
Case 1: Z is connected.
From Proposition 2.3.3, take any x E Co n Exx., x 5 0, for some transitive
orientation 0 of G. From Proposition 2.3.10, we know that x is constant on each
A E P, so i.1 holds. Moreover, since the elements of P are the maximal proper
modules of G, then Lemma 2.3.4 shows that for completely adjacent A, A' E P, xi
xj whenever i E A and j E A', so i.2 holds. Now, since the orientation of GP induced
by x is then equal to the restriction of 0 to the edges of GP, we observe that for A, A'
as above, the edges {{i, j} E E : i E A and j E A'} are oriented in 0 in the same
direction (either from A to A', or vice-versa). Since 0 is transitive, this immediately
implies that its restriction to GP is transitive, so Gp is a comparability graph and i.3
holds. Notably, this holds for any choice of 0. If &is a fiber of x, then we can write
G[ ] as a disjoint union of its connected components, say G[&] = G[B1 ] + - - -+ G[Bkl.
On the one hand, the restriction of 0 to any induced subgraph of G is transitive, so
G[&] is a comparability graph, and also each of its connected components. On the
other hand, from i.2, each Bi with i E [k] satisfies that Bi C A for some A E P, and
moreover, G[Bi] is a connected component of G[A], so Bi is a module G since Bi is a
module of A and A is a module of G. This proves i.4.
As Gp does not have proper non-trivial connected modules, from Theorem 2.3.5
and Corollary 2.3.12, we obtain that dim (EAma.) = 1. Also, Gp has exactly two
transitive orientations and each can be obtained with probability 1 from x E EAmax a
u.c.u.v., proving ii.
Note: In fact, then, it follows that for any x E EAmax\{0}, necessarily x E Co
or x E COduaI, where 0 is the orientation used in the proof, and Odual is the dual
orientation to 0.
Case 2: G is disconnected.
This is precisely the setting of Proposition 2.3.11, so i.1-3 and iii follow after
noting that, firstly, p-partite graphs are comparability graphs, and secondly, their
transitive orientations are exactly the acyclic orientations of their edges such that:
For every pair of maximal independent sets, all the edges between them (or
having endpoints on both sets), are oriented in the same direction.
The proof of i.4 goes exactly as in Case 1.
Corollary 2.3.14. Let G = G([n], E) be a connected comparabilitygraph with canonical partitionP, and let 0 be a transitive orientation of G. Then, (1) the restriction
of 0 to each of G' and G[A], A E P, is transitive.
Conversely, (2) if we select arbitrary transitive orientationsfor each of G' and
G[A], A c P, and then take the union of these, we obtain a transitive orientationfor
G.
Proof. Statement (1) follows from Proposition 2.3.13 and Proposition 2.3.3, since
dim (Co n EAmax) = dim (EAmax)*
47
For (2), select transitive orientations for each of GP and G[A], A c P, and let 0
be the orientation of E so obtained. Since each element of P is independent in GP
and since the restriction of 0 to G' is transitive, then:
(*) For A, A' E P completely adjacent, the edges between A and A' must be oriented
in 0 in the same direction.
This rules out the existence of directed-cycles in 0, so 0 is acyclic. Now, if 0 is not
transitive, then there must exist i, j, k E [n] such that (i, j) and (j,k) are in 0 but not
(i, k). By the choice of 0, it must be the case that exactly two among i, j, k belong
to the same A E P, and the other one to a different A' E P. The former cannot be i
and k, per the argument above (*). Hence, without loss of generality, we can assume
that i, j c A and k E A'. But then, A and A' must be completely adjacent and (i, k)
must exist in 0, so we obtain a contradiction.
Note: The argument for (2) is essentially found in Ramfrez-Alfonsin and Reed
{2001}.
Corollary 2.3.15. Let G = G([n], E) be a connected comparabilitygraph with at least
one proper non-trivial connected module B, and canonical partition P. Then, G has
more than two transitive orientations.
Proof. Suppose, on the contrary, that G has only two transitive orientations. We
will prove that, then, G cannot have proper non-trivial connected modules and so B
does not exist.
From Corollary 2.3.14 and Proposition 2.3.13.ii-iii, a necessary condition for G to
have no more than two transitive orientations is:
(*) G = Gp, and either ? is connected or it has exactly two connected components.
Now, if Z is connected, then B C A for some A c P by Corollary 2.2.7, so B is an
independent set of G since A is independent. This contradicts the choice of B. Also,
if C has two connected components, then G is a complete bipartite graph. However,
it is clear that no such B can exist in a complete bipartite graph.
Proof of Theorem 2.3.1. The different numerals of this result have, for the most
part, already been proved.
- i was proved in Proposition 2.3.3.
- ii was proved in Proposition 2.3.13 for the case when C is connected (See Note).
In the general case, ii follows from Proposition 2.3.13.i.1-3 and Corollary 2.3.14
Statement (2) for x E Ex. a u.c.u.v., and then for all x E Ex, since the cones
Co (with 0 an acyclic orientation of E) are closed.
- iii.1-5 and iv are precisely Proposition 2.3.13.
48
- For iii.6, from Corollary 2.3.15 and Theorem 2.3.5.iii, G has exactly two transitive orientations if and only if G has no proper non-trivial connected modules. Now, if G has no proper non-trivial connected modules, then Proposition 2.3.13.i.4 shows that the fibers of x are independent sets of G and Theorem 2.3.5.ii gives dim (Ex.,,) = 1. Conversely, if the fibers of x are independent sets of G, then G = G'. Furthermore, per Proposition 2.3.13.ii-iii,
if dim (Ex...) = 1, then 0 has at most two connected components. Hence,
G = GP and ? has at most two connected components, so we obtain precisely
the setting of (*) in Corollary 2.3.15. Consequently, G cannot have proper
non-trivial connected modules.
2.4
A characterization of comparability graphs.
This section offers a curious novel characterization of comparability graphs that results from our theory in Section 2.3.
Theorem 2.4.1. Let G = G([n], E) be a simple undirected graph with Laplacian
matrix L, and let I be the n x n identity matrix.
Then, G is a comparability graph if and only if there exists cE R>o and an acyclic
orientation 0 of E, such that CO is invariant under left-multiplication by .I + L.
If G is a comparability graph, the orientations that satisfy the condition are precisely the transitive orientations of G, and we can take Oc = 0 for them.
Proof. If G is a comparability graph and 0 is a transitive orientation of G, then
Step 1 of Proposition 2.3.3 shows that indeed, Lx E Co whenever x E Co. Clearly
then, for all Oc E R>o, (LxI + L)x E CO whenever x E Co.
Suppose now that G is an arbitrary simple graph, and let 0 be an acyclic orientation (of E) that is not a transitive orientation of G. Then, there exist i, j, k E [n]
such that (i, j) and (j, k) are in 0 but not (i, k), and the following set is non-empty:
j
X := {k E [n] : there exist i,
E [n] and directed-edges
(i, j), (j, k) in 0, but (i, k) is not in 0}.
In the partial order on [n] induced by 0, take some f E X maximal, and consider the
principal order filter fv whose unique minimal element is f. The indicator vector of
f' is eev. Then, ejv E Co. Now, choose i, j E [n] so that (i, j) and (j, f) are in 0 but
not (i, f). As f was chosen maximal in X, for every k E fv, k -$ f, then both (i, k)
and (j, k) are in 0. Therefore, we have:
(Leev)i = - |ev| + 1, and
(Leev) 3 = -
fVI
Hence, (Leev)i > (Leev)j and Letv 9 CO since (i, j) is in 0. Since actually eev E &CO,
0
Co for Ct E R>O.
then (cI + L)eev
49
50
Chapter 3
Spanning Trees.
3.1
Introduction.
This chapter is a continuation to Chapter 1, focusing instead on the structural and
enumerative properties of acyclic orientations. It is the most independent and difficult
chapter of the thesis, and even though it maintains the conventions and spirit of the
previous Chapters 1 and 2, we will present it as a self-contained unit of the thesis.
We introduce here a number of novel perspectives, results and resources for the
study and discovery of fundamental properties of acyclic orientations, and their generalization, partial acyclic orientations, of a simple graph; these include polytopal
cell complexes and polynomial ideals {Miller and Sturmfels, 2005}, graphical zono-
topes {Postnikov, 2009; Beck and Robins, 2007}, and Markov chains {Lovisz, 1993;
Aldous and Fill, 2002}, among others. We adopt an original approach to the wellknown connection between labelled trees, parking functions, non-crossing partitions,
and graph orientations. This is the viewpoint of non-crossing trees, not properly
treated or even reported in the literature, and which we exploit to obtain new results
about these objects. Non-crossing trees are, in part, motivated by the techniques
of Fink and Iriarte G. {2010}, but owe their existence to the (subtleties of the)
Alexander duality {Miller, 1998} between two special polynomials ideals defined during Section 3.3 of this chapter. The perspectives presented here complement those
of previous key studies, including but not exhausting, those found in Chebikin and
Pylyavskyy {2005}; Postnikov and Shapiro {2004}; Dochtermann and Sanyal {2012};
Manjunath and Wilmes {2012}; Mohammadi and Shokrieh {2013}; Stanley {1997,
1998}; and the references therein. The present chapter of the thesis is, in fact, an
evident seed for future research more than a conclusive exposition of the topic, and
the number of (sometimes quite provocative) open problems and directions for future
research should gradually become clear.
In principle, an inconvenient aspect of acyclic orientations of a simple graph is
their apparent but, nevertheless, artificial relation to bijective labellings of the vertexset with a totally ordered set. This point of view was exploited during Chapter 1.
Conceivably, adopting a perspective different to that of bijective labellings seems
equally fated to illuminate the study of acyclic orientations of a simple graph, and
51
this is what we pursue during this last part of the thesis. One example of how we
apply ideas developed during this chapter is the construction of a random walk on a
certain simple connected regular graph with vertex-set equal to the set of all acyclic
orientations of any fixed simple graph, which therefore exhibits a unique uniform
stationary distribution. The importance and applicability of such constructions is
evidently exemplified in Broder {1989}; Aldous {1990}; Kelner and Madry {2009};
and formalized in Lovisz and Winkler {1995b,a}. Many other fine works have made
use of similar ideas to solve different combinatorial algorithmic problems.
Understanding acyclic orientations of a simple graph from their grounds usually
entails making precise connections of their theory with the theory of spanning trees,
much better understood; this is also the case in the present chapter. The particular
connection between these sets of combinatorial objects that we choose to follow,
developed here for the first time, is far from obvious and will be presented later in
Section 3.4, where it sprouts naturally from the constructions of Sections 3.2-3.3.
Let us describe in fair detail the contents of the different sections of the chapter.
In Section 3.2 we introduce, again for the first time in the literature, an elegant
inequality description of a well-known polytope related to the acyclic orientations of
a fixed simple (connected) graph on vertex-set [n], n E P; it can be found in Subsection 3.2.1. The above description has the form predicted in Postnikov {2009} for
the generalized permutohedra. This polytope of partial acyclic orientations has a
Minkowski sum decomposition whose summands appear also as summands in Postnikov's expression of the graph associahedron as a sum of simplices. A first step along
this road from the polytope of partial acyclic orientations to the graph associahedron
of graph tubings {Devadoss, 2009} leads us to consider, in the case of connected
graphs, the Minkowski sum of the former polytope with an (n - 1)-dimensional simplex. The construction of Cayley's trick applied to this case serves us to discover
one more polytopal cell complex associated to the graph, a complex pivotal in the
study of certain "artinianizations" of the ideals defined in Section 3.3 and (therefore)
instrumental in the search for minimal free resolutions of these ideals {Bayer and
Sturmfels, 1998}, and whose combinatorial dual is precisely the totally non-negative
part of the graphical arrangement {Stanley, 2004}.
In Section 3.3, this clean geometrical duality of polytopal cell complexes manifests
itself as an algebraic duality between two polynomial ideals associated to a fixed
simple connected graph, defined therein; one of these ideals is motivated by the role
of acyclic orientations in the graphical zonotope, and the other by the inequality
description of the polytope of Subsection 3.2.1. The proof of this Alexander duality,
found early in the section, contains the stepping stones for Section 3.4. We regard
some of the results contained in this section as being "close siblings" to those found
in Dochtermann and Sanyal {2012}, Manjunath and Wilmes {2012} and Mohammadi
and Shokrieh {2013}, yet our modus operandi aims to fix a necessarily problematic (at
least for our purposes) aspect of these other works: The generalization of the duality
between the (standard) permutohedron and tree ideals implicit in them is by no means
self-evident nor truly discussed, and it does not follow from a clean geometrical duality
generalizing the picture of the permutahedron and the barycentric subdivision of the
simplex; as such, these other perspectives do not yield the algorithmic consequences
52
that we need later on in Sections 3.4-3.5.
Section 3.4 introduces non-crossing trees of a simple graph, certain pictorial representations of labelled rooted trees reminiscent of Fink and Iriarte G. {2010}. There
is one non-crossing tree per each rooted spanning forest of the graph. In Subsection 3.4.1, we explain how each non-crossing tree naturally encodes a uniquely determined standard monomial of the generalized tree ideal, defined in Section 3.3, and
(therefore) a uniquely determined orientation of the graph with no directed-cycles.
Among these orientations supported on non-crossing trees, we find the acyclic orientations of the graph, which spring up, again naturally, from non-crossing trees
satisfying a certain efficiency condition. In Subsection 3.4.2, we adopt "the other"
point of view on non-crossing trees, and observe how we then obtain chains of the
non-crossing partitions lattice. These two points of view are combined to produce a
coherent picture of the combinatorial objects involved in this part of the thesis.
Section 3.5 contains applications of the ideas developed in Sections 3.2-3.4 to algorithmic/computational problems involving (mostly random) acyclic orientations.
Subsection 3.5.1 presents five different stochastic processes on state space equal to
the set of all acyclic orientations of a simple graph, and whose stationary distributions
range from dependent on the number of linear extensions (as in Chapter 1) to uniform.
In order of appearance, these are the Card-Shuffling Markov chain, the Edge-Label
Reversal and the Sliding-(n + 1) stochastic processes, the Cover-Reversal random
walk, and the Interval-Reversal random walk. The Card-Shuffling Markov chain had
also been previously discovered as a hyperplane walk in Athanasiadis and Diaconis
{2010}, and the Cover-Reversal random walk is grounded on the work of Savage and
Zhang {1998} and of Section 3.2 of the present chapter. This subsection culminates
with the presentation of the Interval-Reversal random walk, an irreducible reversible
Markov chain with uniform stationary distribution on the acyclic orientations of a
simple graph, never presented before in the literature, and motivated by a close inspection of Section 3.2 here. Subsection 3.5.2 presents a surprising expression for the
expected number of acyclic orientations of an Erd6s-Renyi random graph in terms
of parking functions, a consequence of the study of non-crossing trees in Section 3.4.
Subsection 3.5.3 introduces a commutative-algebraic approach to determining all percolating sets in k-bootstrap percolation on any simple graph, e.g. Balogh et al. {2009};
this direction could yield good fruits if further explored in the future.
3.2
3.2.1
Polytopal complexes for acyclic orientations.
A Classical Polytope.
Definition 3.2.1. Let G = G(V, E) be a simple graph and let:
V 2 =VXV.
An (partial) orientation 0 of G is a function 0 : E -4 V 2 U E such that for all
e = {u,v} E E, we have that O(e) E {e, (u,v), (v,u)}. We will let Otria be the
53
identity map E -+ E.
Definition 3.2.2. For a simple graph G = G(V, E), a partition F_ of the set V is
said to be a connected partition of G if G[a] is connected for all c- E Y, where G[o]
denotes the induced subgraph of G on vertex-set a.
Definition 3.2.3. Let G = G(V, E) be a simple graph and Y a connected partition
of G. Then, the E-partition graph G' = G'(Y-, E') of G is the graph such that, for
o, p c Y with a 4 p, {-, p} E Ey if and only if there exists u E a and v G p with
{u,v} E E.
Definition 3.2.4. Let G = G(V, E) be a simple graph. An orientation0 of G is said
to be a partial acyclic orientation (p.a.o.) of G if 0 can be obtained in the following
way:
There exists a connected partition F of G and an acyclic orientation O of the
E-partition graph G' of G such that, for all e = {u, v} C E:
1. If e - a for some u E I, then O(e) = e.
2. If u C a and v E p for some a, p E E with a $ p, and if O'({-, p}) = (a, p),
then O(e) = (u,v).
We will also consider two functions, dimG and JG, associated to the set of p.a.o.'s
of G. To define them, let 0 be a p.a.o. of G with associated connected partition L.
The first function, dimG, maps from the set of all p.a.o.'s of G to N, and is given
as:
dimG(O) =|Vi -
IT.
The second function, JG, has also domain the p.a.o. 's of G, but it maps to the set
of finite distributive lattices:
JG(O) = J(OQ,
where J(OE) is the poset of order ideals of OF.
Here 0'
is being identified with its induced poset on ground-set Y.
Remark 3.2.5. More generally, for a p.a.o. 0 of G
often identify 0 with its induced partially ordered set
we have that u <0 v if and only if u E a and v C p for
there exist aO, U 1 ,... , ak E I with ao = a and ak = p
as in Definition 3.2.4, we will
(V, :o), where for all u, v E V
some a, p E Y with a f p, and
such that (a-i_1, ai) C 0: [El]
for all i C [k].
Lemma 3.2.6. In Definition 3.2.4, if 0 is a p.a.o. of G, then dimG(O) is equal to
IVi - 1 (JG(O)), where 1(.) denotes the length function for graded posets.
Proof. Let Y be the connected partition of G associated to 0. The result follows
since then 1 (J(0)) = 15-.
Notation 3.2.7. For 0 a p.a.o. of G, JG(O) will denote the ground set of JG(O).
54
Lemma 3.2.8. In Definition 3.2.4, consider a p.a.o. 0 of G, and for I E JG(O), let
IU = IJG 1 u. If we let P be the poset of all I' with I E JG(O), ordered by inclusion
of sets, then P ~ JG(O).
Proof. This is straightforward, since for I1, 12 c JG(O), both ii n I2 E JG(O) and
I1 U 12 E JG(O).
0
Remark 3.2.9. In fact, following Lemma 3.2.8, in Definition 3.2.4 we will regard
JG(-) as a collection of subsets of V ordered by inclusion.
Lemma 3.2.10. In Definition 3.2.4 and Remark 3.2.7, the map
map.
JG
is an injective
Proof. Let 01 and 02 be p.a.o.'s of G such that JG(01) = JG(02). Hence, JG(01) =
Uk = V of this poset, we ob...
JG(02). Considering a maximal chain 0 = uo
of
G associated to both 01 and
partition
serve that Y = {oi\ii}iEkI is the connected
02. The poset of join-irreducibles of JG(01) = JG(02) determines a unique acyclic
orientation OF of the E-partition graph G', and so both 01 and 02 are obtained from
L
0' as in Definition 3.2.4.2. Clearly then 01 = 02.
Definition 3.2.11. Consider a simple graph G = G(V, E). We will define an abstract
, dim,), with underlying set of faces TG ordered by jz,
cell complex _TG = (TG
and with dimension function dime, in the following manner:
1. $G
is the set of p.a.o.'s of G.
2. For 01,02 p.a.o.'s of G, 01 iz 02 if and only if
JG(
0
2) 9
JG(O1).
3. For 0 a p.a.o. of G, dim,(O) = dimG(0).
-
Example 3.2.12. In Figure 3-1, we present two examples of p.a.o.'s, 01 and 02, of a
graph G on vertex-set [15] = {1, 2,..., 15}, such that 02 iz 01. Figure 3-ia shows a
connected simple graph G = G([15, E]). Figure 3-1b presents a particular p.a.o. 01
of G, with associated connected partition E1 (each of its blocks represented by closed
blue regions), and Figure 3-1c the El-partition graph G" and its acyclic orientation
OE'. Similarly, Figure 3-1d shows another p.a.o. 02 of G, with associated connected
partition E2 (blocks represented by closed blue regions), and Figure 3-ie the E 2
partition graph G 2 and its acyclic orientation O2 . Table 3-1f then offers complete
calculations of JG(01), JG(O2), dimG(01) = dim_(01) and dimG(01) = dim,(0 2 ).
Note that since JG(01) C JG(02), then 02
01.
-<
Lemma 3.2.13. Let G = G([n], E) be a simple graph, and let a,b C R and c E R+.
Consider the function F : 2n] -* R given by F(u) = a + blc| + c|E(G[o])|, U E [n].
Then, for all u, p C [n]:
F(u) + F(p)
<;
F(u n p) + F(cr U p).
Equality holds if and only if u\p and p\o are completely non-adjacent sets in G, i.e.
if and only if {{i, j} E E : i E o\p and j c p\u} = 0.
55
a
=j{i =1
1E,
2
,== 11 ,13, 14
13 14, 15},
14,1
G = G([15], E)
1
1
14
(7(72
Q~r)
8
6
4
(a)
(74
(d)
(e)
(b)
(C)
OF2
G-2,
12
GE, OE :
3
92
d4
p. a. o.
01
02
Idime
JG
0, {2, 3, 4, 5, 6}, {1, 2, 3, 4, 5, 6}, {2, 3, 4, 5, 6, 7,8, 9}, {1, 2, 3, 4, 5, 6, 7, 8, 9},
{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 12, 13, 14, 15},
{1, 2,3,4,5,6,7,8,9,10,11,12,13,14,15}
0, {2, 3, 4, 5, 6}, {1, 2, 3, 4, 5, 6}, {2, 3, 4, 5, 6, 7, 8, 9}, {1, 2, 3, 4, 5, 6, 7, 8, 9},
{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 12, 13, 14, 15}, {6}
{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15}
or dim,
10
9
(f)
Figure 3-1: Examples of p.a.o.'s and the order relation
56
2-<
of Definition 3.2.11.
Remark 3.2.14. In standard combinatorial theory terminology, in Lemma 3.2.13,
we say that the function F is lower semi-modular {Crapo and Rota, 1970}.
Theorem 3.2.15. Let G = G([n], E) be a simple graph with abstract cell complex
_TG, as in Definition 3.2.11. Then, the face complex of the polytope:
(3.2.1)
ZG
x
=n+IE and
E R "n]: Exi
iE[n]
xi
> |J- + E(G[o-)I for all u C [n]
iE9
is a polytopal complex realization of
TG-
Proof. Per Lemma 3.2.10 and for the sake of clarity, in this proof we will think of
elements of 2"G as their images under JG.
To begin, an easy verification shows that the point j - dG + 1 lives inside ZG, so
ZG is non-empty. Also, ZG is bounded.
Now, consider a (relatively open) non-empty face F of ZG, and let CF be the
, yj = J-l + IE(G[-])I if y E F.
collection of all u- C [n] such that
A first key step in the proof will be to establish that CF E _TG. We will do this in
a series of sub-steps. Let C' be the poset on ground set CF ordered by inclusion.
Claim i CF is closed under intersections and unions, so C) is a distributive lattice.
Let y E F. By definition, both 0 and [n] belong to CF. Let us now take
yi = |l + IE(G[c-|)I
U, p E CF and let us assume that u V p, p V u. Then, Ei,
so:
and EjEP yj = lpl + IE(G[p])I,
|u U pl+E(G[u U p]) <
yj
iEUP
=ZYitIZYj-LEYk
iEa
jEp
kEaflp
I U pl + E(G[-])I + IE(G[p])I
-
E(G[u- n p])I.
In particular, JE(G[o n p])| + JE(G[c- U p]) !5 jE(G[o-]) + JE(G[p])|. However,
per Lemma 3.2.13:
|E(G[un p])I + jE(G[cr U p])| = |E(G[o])I + jE(G[p])|.
This implies that u n p E CF and U U p E CF-
Claim ii Let 0 = -o C u-1
...
C
,
=
[n] be a maximal chain in CF. Then, G[-IU\o
is connected for all i E [k].
57
1]
Let i c [k] and suppose that G[ji\ri- 1 ] is disconnected. Let pi and P2 be two
completely non-adjacent disjoint sets of G[ui\ui_1] such that Pi U P2 = Ui\0-i.
Then Lemma 3.2.13 shows:
IE(G[uai- U pi])I + lE(G[ui_1 U P21)1 = IE(G[ui])l + |E(G[ui_1])j.
Also, for y E F:
Ioi1 + Iuil + IE(G[a_i U pi]) + jE(G[ui-l U P21)1
= loi- U Pi + IE(G[ui-_ U P1) +
<
E
y +
jEUi-iUpi
=
Ic i-1I +
E
Yk
kEaoi-UP2
oj +
Z
i_1 U p21 + IE(G[uTi U P2])1
Y,+Ey
kEu1
jEri-i
lE(G[Ti_1.]) + IE(G[i])I.
This implies that ui-I U pi E CF and Tiof a maximal chain in CF.
1
U P2 E CF, contradicting the choice
Claim iii For a chain as in Claim ii, suppose that there exist 1, m c oi\c_ 1 with m
i E [k]. If U E CF, then either {m,l} lu = 0 or {m, 1} C a.
$ 1,
Suppose on the contrary that for some o- E CF, m C u but 1 V a. Then,
(a n ui) U
i_
1 E CF per Claim i, and ai.
1 c (- n ui) U a- 1 C ai, which
contradicts the choice of maximal chain.
Claim iv Per Claim iii, every a E CF is a disjoint union of elements of the connected
partition Y := {-i\ui1} [k] of G. Consider the acyclic orientation O of G' =
G'(Y, E) such that for e = {aJi\ai_ 1 , a\aY_ 1}
El and i < j, O(e) =
(ai\Ti1 , a-j\o-j_ 1 ). Then, both E and O are well-defined, i.e. independent of
the choice of maximal chain of C:.
That Y is well-defined follows from Claim i and Claim iii, since distributed
lattices are graded. To prove that O is also consistent, it suffices to check that
if a C p for some a, p E CF such that a and p\- are adjacent in G, then a
set T E CF with NG(a) n (p\-) C r must satisfy that a n NG(p\a) C r. On
the contrary, if o- n NG(p\a) g r, then both T\o- and a\T are non-empty and
adjacent in G. However, since per Claim i Ur E CF, we obtain a contradiction
with Lemma 3.2.13.
Claim v From Claim iv, let 0 be the p.a.o. of G obtained from
a E JG(O).
0
'.
If a E CF, then
This is essentially a corollary to the proof of Claim iv. Consider a maximal
chain- of C that contains a. Then, clearly - c JG(O).
For the first part, it remains to prove that if a E JG(O), then a E CF. This is
easy to establish by considering a point y E F. For p G E, note that Ei
yi =
58
IpI + IE(G[p])
+
IO[E] n ([n]\p x p). But then:
Eyi =
iEU
1: |p + |E(G[p])| + 10[E] n ([n]\p x p)l
pEELp99
= Jul + IE(G[ul)I,
since u E JG(O). Hence, - E CF and CF = JG(O) E _c.
For the second part, let us take a JG(O) E _T' for some p.a.o. 0 of G, and we
want to prove that JG(O) = CF for some (relatively-open) non-empty face F of ZGThe first part gives us a clear hint of how to proceed here. Let us consider the point
y E
nR]
given by yz = IO[E] n ([n]\{i} x {i})| + 1 -IO[E] n {e c E: i E e}|+ 1 for
all i E [n]. Since 0 is a p.a.o. of G, then for a E JG(Q), we have that EiEy
=
0.
On
the
U
{{i,j}
:
i
E
u,j
E
[n]\-})
=
l- + |E(G[oi)|, as O[E] n (([n]\o x u)
iEU Yi > Jul + IE(G[u])I.
contrary, if u V JG(O), the later set is non-empty and
Therefore, JG(O) = CF for some face F of ZG.
We have now established how _TG corresponds to the set of (relatively-open) nonempty faces of ZG. Naturally, i, corresponds to face containment and dim, to
affine dimension, and the correctness of these two is an immediate consequence of our
correspondence and of basic properties of the inequality description of a polytope. 0
G([n], E) be a simple graph with graphical zonotope
and degree vector dG, where (see Definition 3.2.19 for notation):
Corollary 3.2.16. Let G
Z
ntra
=
Zg";"
[ei - ejej
:=a
-e].
{i,j}EE
Then:
ZG={
2
1
Zentral
+
1
-
2
' dG
+ 1-
(3.2.2)
Proof. As it is known from Chapter 1, the vertices of Zy"n"' are given by all points
of the form x 0 = (indeg(GO)(i) - outdeg(G,0 )(i)) E[]I where 0 is an acyclic orientation
"a"t by 1 - dG + 1, we obtain that the vertices of the
new polytope are given by all vectors of the form yo = (indeg(GO)(i) + 1)iE nl, where
L
o is an acyclic orientation of G, but these are precisely the vertices of ZG-
of G. Hence, translating 1
.
Zjn
Definition 3.2.17. Let G = G([n], E) be a simple graph with graphical zonotope
Zntral. From Corollary 3.2.16, we will call the polytope ZG the clean graphical zonotope of G.
3.2.2
One More Degree of Freedom.
Definition 3.2.18. Let G = GC([n], E) be a connected graph. Define %YG = (_G, jy, dimy
to be the abstract cell complex with underlying set of cells 3YG, order relation -<y, and
dimension map dimy, given by:
59
_G
-
XG U
I
(2 [n]\{n],0}), where:
G:
{(cr, O) : 0
#
a
[n], and O is a p.a.o. of G[]}
.
1.
2. For A, B c _G with A # B, we have that A -y
following holds:
B if and only if one of the
(a) If A, BE 2[n]\{I[n],0}, then A C B.
(b) If A E 2 []\{[n],0} and B
=
(Y,O) E XG, then A C [n] \u.
(c) If A = (uo, Oo), B = (ci, 0 1 )
c
XG, then JG[ai](01) C JG[uo](Oo).
3. For A E YG:
(a) If A E 2Ln\{[n],0}, then dimy(A) = JAl - 1.
(b) If A = (ulO) c _XG, then dimy(A) = l[n]\oi+dimG[-](O).
Definition 3.2.19. Let S and T be non-empty subsets of RI[n. The join of S and T
is the set:
[S, T] := {x E Rin] : x = xs + (1 - a)t for some ot c [0, 1], s E S and t C T}.
The strict join of S and T is the set:
(S, T) := {x E R[n] : x = cts + (1 - cx)t for some 0C E (0, 1), s C S and t E T}.
Proposition 3.2.20. Let P and Q be (n - 1)-dimensional polytopes in R' such that
aff (P) and aff (Q) are parallel and disjoint affine hyperplanes. Consider an open
segment (x, z) with x E P, z E Q, and let y C [P, Q]. Then, the following are true:
ii (x, z) 9 1 ([P, Q]) if and only if for every p E relint (P) and e > 0, z + e(x -p)
Q.
On the contrary, (x, z) 9 int ([P,
and F > 0, such that z + (x - p) E
Q])
(
i y E int ([P, Q]) if and only if there exist p* E relint (P) and q* E Q such that
y E (p*, q*), if and only if there exist p** C relint (P) and q** C relint (Q) such
that y E (p**,q**).
if and only if there exists p E relint (P)
Q.
iii Let 7 arff(P) : R n) 4 R[n] be the projection operator onto the affine hyperplane
containing P. If taff(P)[(x, z)]n relint (P)n relint (aff(P) [
0, then (x, z) G
int ([P, QI).
Proof. We will obtain these results in order.
i (See also Figure 3-2b) We prove the "only if" direction for both statements.
Suppose that y E int ([P, Q]) and let p C P and q c Q be such that y E (p, q).
Let us assume that p E a (P). Take an open (n - 1)-dimensional ball By C
int ([P, Q]) centered at y such that aff(By) is parallel to aff(P) and aff(Q). Let
60
C be the positive open cone generated by By - q, and consider the affine open
cone q + C. Then, B,, : (q + C) n aff(P) is an open (n - 1)-dimensional ball
in aff(P) such that p E relint (B.). Hence, since P is also (n - 1)-dimensional,
there exists some pi E relint (P) n B.. Now, let yi = (p1, q) n By E int ([P, Q]).
Since Y2 := y + (y - yi) E B, 9 int ([P, Q]), there exist P2 E P and q2 E Q
such that Y2 = (p2 , q2) n BY. But then, there exist p* E (P1, P2) C relint (P)
and q* E (q, q2 ) g Q such that y = (*, q*) nBy, as we wanted. If q* E a (Q),
we can now repeat an analogous construction starting from q* and p* to find
p** E relint (P) and q** E relint (Q) such that y E (p**, q**).
ii This is a consequence of i, and not easy to prove without it. We prove the second
statement, which is equivalent to the first. For the "if" direction, suppose that
for some p E relint (P) and e > 0, z + F(x - p) E Q. Take some y E (x, z) and
consider the line containing both z + e(x - p) and y. For a sufficiently small e,
this line intersects aff(P) in some pi E relint (P). But then, for a small open
ball Bp, C relint (P) centered at p, and with aff(Bp 1 ) = aff(P), the open set
(Bp 1 , z) contains y and lies completely inside int ([P, Q]), so y E int ([P, Q]).
For the "only if" direction, suppose that (x, z) C int ([P, Q]) and take y E (x, z).
If x E relint (P), then we are done since Q is also (n - 1)-dimensional. If
x c a (P), from i, take p c relint (P), p # x, and q E Q with y E (p, q). But
then, z + E(x - p) = q E Q for some e > 0.
iii Take p E flafP) [(x, z)] n relint (P) n relint (7rtaffLp) [Q]) and let p = 0 be a normal
to aff(P). Then, for some y E (x, z) and real number Oc : 0, y E (p, p + Ocp)
and p + Ocp E relint (Q), so i shows that y E int ([P, Q]). Clearly then (x, z) C
int ([P, Q]).
,
Definition 3.2.21. Let G = G([n], E) be a simple graph, and let 0 be a p.a.o. of G
with connected partition Y and acyclic orientation OF- of G'. Let us write Y0,in for
the set of elements of E that are minimal in (E, o5 ), and for i E [n] with i E p E
let:
IG(i, O) ={- E Z: a <or p}, and
IG(i, O) ={j E [n] j E u E Y and a >or p}.
With this notation, we now define certain functions associated to 0 and G, called
height and depth:
height , depthG :
height(i)
[n]
-*
Q,
1
-
depthG(2) =
height(j).
jEIV(i,O)
Example 3.2.22. Figure 3-2a exemplifies Definition 3.2.21 on a particular graph
G on vertex-set [14] = {1, 2,... ,14}, with given p.a.o. 0. Since both heights
61
aff(P
0:
I
a 614414
'
5-"
b)aff
1 3
E ={0,U2,
3},/p=
heightg constant on each
{117, 1}
ci. i E
depthG constant on each oa, i
E
161, e.g. height (6)
[61, e.g. depthG(13)
=
1
=
(a)
(b)
Figure 3-2: Visual aids/guides to the proofs of (3-2a) Proposition 3.2.23 and (3-2b)
Proposition 3.2.20.i. (3-2a) also offers an example for Definition 3.2.21.
are constant within each element/block of the connected partition E =
{oa = {7}, -2 = {1, 2}, 0 3 = {6,10,14}, .., o-6 = {3, 4, 8}} associated to 0, we present
only that common value for each block in the figure.
and
depth'
1 >
depthG(i) >
.
Proposition 3.2.23. In Definition 3.2.21, let 9 E JG(O) and let p 9 [n] intersect
every element of 5i~ni in exactly one point and contain only minimal elements of 0.
Then:
iEpn(T
if
and only if
= [n], and
.
Moreover, if G is connected, then iE np ePtho(i) >
n depthGi) - E>
whenever this holds,
Remark 3.2.24. Figure 3-2a shows one such choice of a set p in Proposition 3.2.23
that works for Example 3.2.22 (in red).
Proof. The verification is actually a simple double-counting argument using the fact
that o- is an order ideal, so we omit it. When G is connected, if o 5 [n], then there
must exist i E [n]\ that is strictly greater in 0 than some element of u (and hence
strictly greater than some element of p), again since u is an order ideal. Clearly, we
must have heightg(i) >
-
Theorem 3.2.25. Let G = G([n], E) be a connected simple graph with abstract cell
complex G as in Definition 3.2.18. For N > 0, N / n + E|, consider the (n
1)-dimensional simplex NA = cony (Nei, Ne 2 ,.... Nen) in R "n.If we let YG be
62
the polytopal complex obtained from the join [ZG, NA] after removing the (open) ndimensional cell and the (relatively open) (n - 1)-dimensional cell corresponding to
NA, then YG is a polytopal complex realization of g(G.
Proof. Let the faces of 3G obtained from 2H'1\{1 n1,o correspond to the faces of 0 (NA)
in the natural way. Also, let the faces of XG of the form ([n], 0) correspond to the
faces of ZG as in Theorem 3.2.15. The result is clearly true for the restriction to this
two sub-complexes, so we will concentrate our efforts on the remaining cases.
First, for the sake of having a lighter notation during the proof, we will let ' =
[n]\p for any set p g [n].
A (relatively open) cell of YG\(ZG U a (NA)) can only be obtained as the strict
join of a cell of 0 (ZG) and a cell of a (NA), so let us adopt some conventions to refer
to this objects.
Convention 3.2.26. During the course of the proof, we will let S (or So) denote a
generic non-empty relatively open cell of NA obtained from p 9 [n] (resp. po), and F
(or Fo) a generic relatively open cell of ZG with p. a. o. 0 of G, associated connected
partition Y of G, and acyclic orientation 0 ' of G' yielding 0 (resp. 00, Eo, 0o1o).
We argue that we will be done if we can prove the following claim:
Claim i a) (F, S) is a cell of YG if and only if b) p = [n] and p is a non-empty union of
elements from the set {u E E: u is maximal in (1, oz)}. When this equivalence is established, then we will let (F, S) correspond to the pair (', O3) E XG,
where O1i denotes the restriction of 0 to E(G[]).
Indeed, assume that Claim i holds. Then, under the stated correspondence of
ground sets of cells, all elements of XG are uniquely accounted for as cells of YG. This
is true for ZG clearly, and for the remaining cases since for any choice of a1 G [n],
5 0, and of p.a.o. 01 of G[u1], we can always extend uniquely 01 to a p.a.o. of G
$in which all the elements of 'ij are maximal.
Secondly, we verify that -< corresponds to face containment in 9 G. Suppose that
(Fo, So) and (F, S) are cells of YG. Then, (Fo, So) 9 (F, S) if and only if Fo g F and
So g 5, if and only if JG(O) C JG(OO) and po 9 p. Now, assuming Claim i, the last
statement is true if and only if JG[p](OI) 9 JG[ -](0j):
The difficult part here is the "if" direction. Clearly, po g p. Since p is a union
of elements of Y that are maximal in (1, <0z), then JG(O) \JG[](0I1) consists
of ideals of 0 whose intersection with p are non-empty unions of the connected
components of G[p]. But then, as p E JGj(015) G JG1as (Oolpo) 9 JG(Oo),
these must also be ideals of JG(Oo).
The analogous verification pertaining to faces in XG of the form ([n], O), or corresponding to Definition 3.2.18.2a-2b, is now a straightforward application of the same
ideas, so we omit it here.
The correctness of dimy will be established in Claim i.3, so indeed if Claim i
holds, the statement of the Theorem follows.
Let us now begin with our proof of Claim i, which consists of three main steps.
63
Claim i.1 Let F and S satisfy the conditions of Claim i.b). Then:
(F, S)
C
([ZG, NA]).
Let x c F and z C S. We must have that 0
#
0
rivial
here.
Now, since G
is connected, there exists o- E Y that is minimal but not maximal in (I, or).
Hence, u n p = 0 and moreover, u C JG(O). But then, by the inequality
description of ZG, for any p E relint(ZG), EEU Pi > Icl + IE(G o])I= E= xI
and x - p must have a negative entry in a. Therefore, z + (x - p)
all E > 0 and Proposition 3.2.20.ii shows that (x, z) C 0 ([ZG, NA]).
'
NA for
Claim i.2 Let F0 , 00, 1o, 0", So, po be as in Convention 3.2.26. Then, there exist F, 0,
T, O', S, p also as in Convention 3.2.26, such that p is a union of elements of
the set {0 E Y : ux is maximal in (T, oY)} and (Fo, So) 9 (F, S).
(See Figures 3-3a and 3-3b for a particular example of the objects and setting
considered during this proof) Let:
o'po :={xE Y-0 : If g E Eo and g
Q:o
u, then g n po =0
Then, define:
c 0 :=
U (.
+ G[Uk] is the decomposition of G[6'O] into its connected
components, we will let E = Xo,p, U {C-, ... , Uk}. We will use the acyclic
orientation O of G' obtained from the two conditions 1) OEroK
=
Oo
and 2) 71, - - I, l are maximal in (E, oz). The p.a.o. 0 is now obtained from
OE, and let F be associated to 0 and S be obtained from ' =1 U ... U UkWe now prove that (Fo, So) C (F, S).
if G[& 0'] = Gla1] + -
Since (Fo, So) C (F, S), it is enough to find x E F and z E So such that
(x. z) C (F, S), so this is precisely what we will do. To begin, we note that for
i E [k], the restriction Oi := Ool, is a p.a.o. of Gi := G[x], so we will let i be
the connected partition of Gi and Of' the acyclic orientation of G' associated
to Oi; moreover, we note that po intersects every element of i minimal in
(Is, 5z). Hence, let us select go C po so that for every i E [k], go intersects
every element of i minimal in (Yi,
i) in exactly one point and so that go n -i
contains only minimal elements in Oi. Now, take any x c F and let:
z
= E
E
depth
(j) -
ej
S.
iE[k] iEeoni
We will make use of the technique of Proposition 3.2.20.ii to prove that (x, z) E
64
00:
Eo = {{1, 15}, {2, 18}, {3, 6, 7, 16}, {4, 13, 17},
{5,9,12}, {8}, {1
0:
E
11},{14}}
{{1, 15}, {2, 3, 6, 7, 8, 16, 18}, {4, 13, 17},
{5,9, 12}, {10, 11}, {14}}
15
po = {1, 2 ,3 ,16 }
w
Eo,p, = {{4, 13, 17}, {5, 9, 12}, {14}, {10, 11}}
0o = {4, 5, 9, 10, 11, 12, 13, 14, 17}
E=
01
{{1, 15}},
{1, 15}, 02
{{2,18}, {3,6,7,16}, {8}}
{2, 3,6,7,8, 16, 18}
2 =
=
(b)
(a)
Figure 3-3: An example to the proof of Claim i.2 in Theorem 3.2.25.
(F, S), so for that we need to consider a point in S, which we select as:
S=s=LN
X1~
E
iE[k]
S
ej (E S.
jEi
For i E [k], if we consider a pi E JG,(O) with pi , a-, Proposition 3.2.23 gives
us:
)
depthQ
d
j Qi
>N (eoi
k jai
--
k
ji
/
\jEeingo
1
Ii12
lid
Jai I
_
N
k.IcrI| 2
> 0.
Hence, for a sufficiently small e > 0, x + E(z - s) E F, so for each y E (x, z) we
can find x' E F and s' E S such that y E (X', z'). That implies (FO, So) ; (F, S).
Claim i.3 Let both F, S and FO, So satisfy the conditions of Claim i.b). Then, (F, S) n
#
0 if and only if F = FO and S = So. Moreover, (F, S) is a face of YG
and dimff ((F, S)) = IpI + dimG[I(01) (similarly for (F,So)).
Let O E (0, 1) and consider the polytope P, = {x E R[n] : i[]xi =
El) + (1 - cx)N} n [ZG, NA]. Every x E P, satisfies the inequalities E
(n
+
(Fo, So)
Xi =
(1 - a)N + oc(n + JEJ) and Eje, xi ;> oc(lu-|+ E(G[u])I) for all u- C [n], a- =' 0.
Per Claim i.1 and Claim i.2, the set (F, s) n P, can be characterized by the
condition that it contains all the points x E P, which, among those inequalities,
65
satisfy the and only the following equalities:
Z xi = (1 -
(3.2.3)
c)N + cx(n + El) and
iE[n]
(3.2.4)
Exi = OC(jj + E(G[u-)),
iE9
for all U E JG[p](Olp), o
$
0.
This observation proves the first statement.
For the second statement, we assume without loss of generality that N > n+ El
and select generic coefficients 0, E R+ with U E JG[p](Ojp)\{0}, such that:
B:,(ju +
IE(G[u])I)
=
N - (n + El).
-E JG[p](O1- )\{O}
The linear functional,
f:=
Ze
iE[n]
+
E
UEJG[O](Oji)\{0}
f.Ze,
(3.2.5)
jEU
satisfies that, for x E P,
f(x) ;> (1 - oc)N + cv(n + El) + c (N - (n + |El))
=
N.
By the proof of the first claim, this inequality is tight if and only if x C
(F, S) n P, = (F, S) n Po. Moreover, since this minimum is independent of
c, the linear functional f is minimized in [ZG, NA] exactly at (F, S). If N <
n + IEl, we must select negative coefficients and consider instead the maximum
of the linear functional in question, analogously.
For the third statement, we simply note that an open ball in the affine space
determined by all x E R "n satisfying Equalities 3.2.3-3.2.4 can be easily (but
tediously) found inside (F, S). Hence, dim,,. ((F, S)) = II + dimG[ ](OI).
Definition 3.2.27. Let G = G([n], E) be a connected simple graph. Let X% =
(XG, -<,dimx) be the abstract cell complex dual to XG in Definition 3.2.18. Hence,
for all (Uo, O), (U-1, 01) E XG:
1. (goo00) 3. (U 1 , 01 ) if and only if JG[ao](Oo) C JG[u 1 ](01), and
2. dimx(o-o, Oo) = l-ol - 1 - dimG[uo](Oo).
Theorem 3.2.28. Let G = G([n], E) be a connected simple graph with abstract cell
complex X as in Definition 3.2.27. Then, the polytopal complex XG whose (closed)
faces are the intersections F n S with F E F(AG) and S a face of (the complex) A
66
is a polytopal complex realization of X-, where F(AG) is the complete fan in R.1n
the graphical arrangement AG of G and A = conv (ei, e 2 , ... ,en):
AG := {Ix E Rn] : X _j
of
= 0 , for some {i, j} G E}.
Proof. From Theorems 3.2.15-3.2.25, and letting N -+ oo in Equation 3.2.5, we know
that the relatively open cone C+o in the totally non-negative part of the normal fan
of the polytope [ZG, NA] that corresponds to a cell (u, 0) E _G, is given by:
C
= spanR+
_e : p E JG[,](O)\jo
iEP
.
+
Hence, since the affine dimension of the corresponding dual cell in YG is [n]\oj
dimG[,](O), then dimff(C+,o)) = n - [n]\uI + dimG[o](O) = Jl - dimG[,](O) and so
dimff (Cg o) n A) = |ul - 1 - dimG[ 0.J(O), since C+) C span 0 {ei, . . .,
gentially, we can also express C+o) more compactly by means of its positive basis as:
C+'o
=
spanR+ {
e ej : p E JGj,](O)\{o>and G[p] is connected}.
Now, the intersection,
F(AG[,])+ :={Fn {x C R '] : xi = 0 if i E [n]\u, and xi > 0 if i E 9}
: F E F(AG)}
is, as suggested by our choice of notation, equal to the totally positive part of the fan
of the graphical arrangement of G[o], regarding here R' as a subspace of RHn1. Per
Theorem 3.2.15, since F(AG[,]) is precisely the normal fan of ZG], and F(AG[])+ the
totally positive part of this fan, we know that the relatively open cones of F(AG[,))
correspond to the p.a.o.'s of G[a]. From the description of the cells of ZG[,], the cone
C+, is exactly the cone in F(AG[,])+ normal to the cell of ZG[Y] corresponding to 0.
This establishes the correspondence between cells of XG and elements of XG, since
we can go both ways in this discussion.
Using this same lens to regard cells of XG, the correctness of Definition 3.2.27.1
now follows from the analogous verification done in Theorem 3.2.25, by a standard
result on normal fans of polytopes, namely, the duality of face containment.
3.3
Two ideals for acyclic orientations.
Definition 3.3.1. Let G = G([n, E) be a simple graph.
67
1. For an orientation 0 of G and for every i E [n], let:
j E [n]}I
Outdeg(G,1O)
11ij E
nod(G,O)(i)
,
indeg(G,O)(i) := {(j, i) E O[E]
[]jE[
7I
I{e E O[E] : either e = (j, i) or e = {i, j}, j E [nG}|,
where we denote the respective associated vectors in R["I as indeg(0 0
and nod(c,o).
2. For a C [n] with a $ 0, define 1
Let now, for every i E [n]:
=
inof(G,,) (i)
:=
0
:= 1[n].
if i C o-,
otherwise,
I{{i,j} E E :
Outof(Ga)()
outdeg(G,O),
ei,e E RNI, further writing 1
E E :j E a}
f{{ij}
),
[n]\}
0
if i E a,
otherwise,
and denote the respective associated vectors of R [n] as inof(G,,) and outof(0,0).
Remark 3.3.2. During this section, we will follow the notation and definitions
of Miller and Sturmfels {2005}, Chapters 1,4,5,6 and 8, in particular, those pertaining
to labelled polytopal cell complexes. We refer the reader to this standard reference on
the subject for further details. Some key conventions worth mentioning here are:
1. The letter k will denote an infinite field.
2. For a := (ai, a2 ,.
. . , a,)
E NHn, ma := (x
: i
C [n]) is the ideal of k[x1,.
..
, xn]
associated to a.
Definition 3.3.3. Let G = G([n], E) be a connected simple graph. The ideal AG of
acyclic orientations of G is the monomial ideal of k[x1,... ,xn] minimally generated
as:
AG :-
Xindei(GO)
J
: 0 is an acyclic orientation of G).
indeg(Go)(i)+l
iE[nI
Definition 3.3.4. Let G = G([n], E) be a connected simple graph. The tree ideal TG
of G is the monomial ideal of k[x1,..., x.] minimally generated as:
TG :=
Xoutof(G,a)1c
x
utof(G,)(0+1
:9 E
2 [nI\{o}
and G[3-1 is connected)
t 3iEoc
Definition 3.3.5. Given two vectors a, b E
68
N En]
with b -- a (bi < ai
for all i E [n]),
let a\b be the vector whose i-th coordinate is:
ai+ 1 - bi
0
ai\bi
ar~i
=
1,
if bi
if bi = 0.
If I is a monomial ideal whose minimal generators all divide x", then the Alexander
dual of I with respect to a is:
Ial := fl{ma\b
Xb
is a minimal generator of I}.
Theorem 3.3.6. Let G = G([n], E) be a simple connected graph. Then, the ideals
AG and TG of Definitions 3.3.3-3.3.4 are Alexander dual to each other with respect to
dG + 1, So AdG+1] = TG and T [dG+l
=
AG-
Proof. It is enough to prove one of these two equalities, so we will prove that
A[dG 11 - TG. Take some o E 2["l\{o} such that G[o] is connected and consider
Utof(G,fJ)M1. We will
the minimal generator of TG given by XOUtOf(G,)+ l =
verify that x"Utf(G,g)1' E m(dG+I)\b for every minimal generator Xb of AG. Se(G,)
HE[] i
lect an acyclic orientation 0 of G and let xindg(G,o)
the minimal generator of AG associated to 0. If we take m E u to be maximal
in ([n], <o) among all elements of a, so that i >0 m and i E a imply i = m,
then ( dG(m)
outof(GM)(m)
+
)
\
outdeg(GO)(m) + 1 < ING(m)\oi + 1
(indeg(GO)(M) + )
+ 1. Hence,
Xoutof(G,+1,
E
OtOf(GU)(m)+1) C KdG(m)+1)\(inde9(GO)(m)+1)
(dG+1A(indeg(G,O)
1)
This proves that TG C A[dG+l]
Now, consider a monomial x' V TG with 0 -< b (so bi > 0 for some i E [n]).
Then, for every U E 2" \{0 } there exists i E - such that bi < outof(G,cr)(i) + 1,
noting here that the condition on G[o] being connected can be dropped. Hence,
consider a bijective labeling f : [n] -+ [n] of the vertices of G such that bf-1(j) <
outof(G,f-Il{,)(f(i) + 1 for all i E [n]. If we let 0 be the acyclic orientation of
G such that for every e = {i,j} E E, O(e) = (ij) if and only if f(i) < f(j),
then for all i E [n], bf-1(j) < outof(G'f-(1,i])(@) + 1 = outdeg(GO)(f-(i)) + 1 =
(dG(f'(i) + 1) \ (indeg(G,O)(fr(i)) + I
therefore TG A [dG+1I
that xb O A [dG+l,
GG
,
or xb
'
M(dG+1)\(indeg(G,O)+1).
Corollary 3.3.7. Let G = G([n], E) be a simple connected graph. Then:
AG
{minof(Ga)+1
a
E 2 [n]\{0} and G[-] is connected},
69
This shows
is the irreducible decomposition of AG. Also:
TG =
{moutdeg(0,G)+1 : 0 is an acyclic orientation of G},
is the irreducible decomposition of TG.
Definition 3.3.8. For a simple connected graph G = G([n], E), consider the polytopal
complexes ZG, YG and XG, which respectively realize the abstract cell complexes _TG,
G and X
of Definitions 3.2.11, 3.2.18 and 3.2.27. We will let ZG = (ZGz
(YG, fy) and XG = (XG, x) be the Nin] -labelled cell complexes with underlying
YG
polytopal complexes given by ZG, YG and XG, respectively, and face labelling functions
Ze 1 , EX, defined according to:
1. ZG: For a face F of ZG correspondingto 0 E 2G:
[n].
fz(F)i = nod(G,O)(i) + 1, i
2. YG:
(a) For a face F of YG correspondingto (u0,O) E
f(F)
-
fy(F~i
G:
+ 1 if i E u,
nod(G[a],o)()
dG(i) + 2
(b) For a face F of YG corresponding to u-
G
otherwise.
C2
dG(i) + 2
0
{]\{[n],0} g 3G:
if i Co,
otherwise.
3. XG: For a face F of XG corresponding to (,-,O)
Ec
:
( outdeg(G[U], 0 )(i + outof(G,,(i) + 1
fF.0
otherwise.
if
E,
Lemma 3.3.9. Let G = G([n], E) be a simple connected graph. Then, for any face
F of ZG with vertices v 1,... , V, we have that:
Xez(F) = LCM {x z(vi)}iE[k]
where LCM stands for "least common multiple".
Proof. Let F be a face of ZG with corresponding p. a. o. 0 of G and connected
partition Y. Every acyclic orientation of G that corresponds to a vertex of F is
obtained by 1) selecting an acyclic orientation for each of the G[o] with u CE, and
then by 2) combining those III acyclic orientations with 0[E] n V 2 . For a fixed vertex
i c o with u c 5, it is possible to select an acyclic orientation of G[-] in which i
70
is maximal and then to extend this to an acyclic orientation of G that refines 0, so
if vertex v, of F corresponds to one such orientation, then e,(vj)j = nod(G,o)(i) + 1.
On the other hand, clearly Ez(vj)i 5 nod(G,o)(i) + 1 for all vertices vj of F. Hence,
xez(F) = L CM{xfz (v)}Elk]
Corollary 3.3.10. Similarly, for G as in Lemma 3.3.9 and for any face F of YG
with vertices v 1 ,... ,V, we have that:
X y(F) = LCM {x 4 (vi)}iE[k]
where LCM stands for "least common multiple".
Proof. If F is a face of YG inside the simplex NA, then this is immediate. If F
corresponds to some (u, 0), then this is a consequence of the proof of Lemma 3.3.9,
since the vertices of F are all the N - ej with i E [n]\c-, and all the vertices of YG
that correspond to acyclic orientations of G whose restrictions to G[u] refine 0 and
in which all edges of G connecting u with [n]\u are directed out of u.
Proposition 3.3.11. Let G = G([nl, E) be a simple connected graph. The cellular
free complex FyG supported on YG is a minimal free resolution of the artinianquotient
k~xi, . . . , x.]/ (AG + mdG+2).
Proof. Without loss of generality, we assume here that N > n + IEj. From standard
results in topological combinatorics it is easy to see that for b E N1"I, the closed
b} form a
faces of YG that are contained in the closed cone C-s = {v E R in) :v
contractible polytopal complex, whenever this cone contains at least one face of YG.
Now, suppose that b satisfies that bi 5 dG(i) + 1 for all i E [n]. Then, the complex
of faces of YG in the cone Csb coincides with YG,-<b, so the later is contractible and
acyclic if non-empty. On the contrary, let Ub be the set of all i such that bi ;> dG(i) + 2,
and let Db = [n]\Ub. Consider the vector a e R["] such that:
-
N
-{
ai= bi
if i E Ub,
if i E Db.
Then, the set of faces of YG in the cone C-<a coincides with YG,-<b, so again the later
is contractible and acyclic if non-empty. This shows that FyG supports a cellular
resolution of k[x 1 , . . . , xn]/ (AG + mdG+ 2 ).
To prove that this resolution is minimal, it suffices to check that whenever Fo and
F 1 are closed faces of YG such that Fo C F1 , then ey(FO) -< y(F). There are three
cases to study:
1. Fo and F 1 correspond respectively to -o, 91 E 2 "]\{[n], 0 } g
9_:
Then, -o C u- and for i E u1 \o, fy(Fo)i = 0 < dG(i) + 2 =y(F).
and F to (- 1 , 01) E
and for i E ai, ey(Fo)i = 0 < 1 < nod(G[o)(i)+1
2. Fo corresponds to -o E 2[n]\{[n],0}
Then, -o 9 [n]\-
1
9 O
71
__:
=y(F).
3. F and F correspond respectively to (o, 00), (- 1 , 01) E X3:
Therefore, JG[t1 I(01) C JG[u](Oo), so 1) if -1 C o, then for i c a 0 \a~1 ,
y(Fo) =
y(F1); and 2) if o-= -o = ui, then
letting Eo and L7 be the connected partitions of G[a] corresponding respectively
to 00 and 01, we observe that 1o is a strict refinement of 11, so there exist
po E Lo and pi E L7 such that po C pi and such that for some i c p,\po, we
f(F 1 )i, since G[p1 ]
have that fy(F6)j = nod(Gao ,O0 ) + 1 < nod(G[i],Ol)(iy1
is connected (so there is an edge directed out of i in Oo which was not directed
in 01).
nod(G[ 3 0I,o0 (i)+I< dG(i) + 1 <dG(i)+ 2 =
-
Proposition 3.3.12. For G as in Proposition3.3.11, the cellularfree complex TzG =
YG,3dG+1 supported on ZG gives a minimal free resolution of the quotient ring:
k[xi , ... , Xn]/AG
Proof. This follows from the proof of Proposition 3.3.11, since ZG
=
G,dG41'
Corollary 3.3.13. For G as in Proposition 3.3.11, let YJ
-' dG + 2 -YG.
Then,
the cocellular free complex F G,dG+1 supported on Y.GdG 1 is a minimal cocellular
resolution of the monomial ideal TG.
Proposition 3.3.14. For G as in Proposition 3.3.11, the cellular free complex TxG
supported on XG is a minimal cellular resolution of the monomial ideal TG.
Proof. This is now a consequence of Corollary 3.3.13, since the underlying polytopal
complex of Y6o-dG 1 iS Combinatorially dual to the underlying complex of XG, and
cells from both complexes dual to each other have equal labels: If a face FY of YG
and a face F, of XG both correspond to (a, 0) c XG, then,
dG
(F) =
_
f
dG(i)+ 2 - (nod(G[a],O)(i) + 1)
dG) 2 - (dG (i) + 2)
outdeg(G[]O)(i) + outof(G,-)(i + 1
0
=
if i E U,
otherwise,
if i E U,
otherwise.
ex(Fx).
The following is, in reality, a well-known result about Betti numbers of monomial
quotients with a given cellular resolution, and not a definition. We present it here
as a definition given its immediate connection to the topology of cellular complexes,
clearly central for the results of this section.
72
Definition 3.3.15. If Fx is a cellular resolution of the monomial quotient S/I, then
the Betti numbers of I are the numbers calculated, for all i > 1, as:
P@i,b(I) = dimk li_1(Xb; k),
where H. stands for the reduced homology functor.
Lemma 3.3.16. For a simple connected graph G = G([n], E), the Betti numbers of
the ideals AG and TG satisfy that, for all i > 0:
Z
Pi,b(AG) =
# p.a.o's of G on n - i connected parts,
bEN[n]
s 1i,b(TG)
=
# of pairs (0, u) : 0 is a p.a.o. of G[-] on i + 1 connected parts.
bEN[n]
Proof. These results are clear from our choice of minimal cellular resolutions for
these ideals, since i-th syzygies of each ideal correspond to i-dimensional faces of the
respective geometrical complex.
0
3.4
Non-crossing trees.
In this section we investigate, for a simple graph G = G(fn], E), a useful and novel
unifying relation between the standard monomials of TG, the rooted spanning forests
of G, and the maximal chains of the poset of non-crossing partitions. We show
that, arguably, the phenomenology that binds these objects together and which has
been hitherto discovered in the literature, is largely due to the existence of a simple
canonical way to represent rooted spanning forests of a graph on vertex-set [n] as
non-crossing spanning trees.
An analogous extension of the theory presented here to a more general poset of
non-crossing partitions associated to G, and the consideration of the equally arbitrary non-nesting trees and their connection to the Catalan arrangement, will not be
discussed here, and will be the subject of a future writing by the author not included
in his thesis.
Definition 3.4.1. For a simple graph G = G(V, E), we will let Gr denote the graph
on vertex-set V U {r} and with edge-set E Li {{r,v} : v E V}, so Gr is the graph
obtainedfrom G by adding a new vertex r and connecting it to all other vertices in G
(e.g. Figures 3-5a and 3-5b).
Definition 3.4.2. A planar depiction (D, p) of a finite acyclic di-graph T = T(V, E)
is a finite union of closed curves D C R 2 and a bijection p : V -+ {0, 1, 2,... ,V
(called a depiction function) such that:
1) p is order-reversing, so if e G E and e = (u, v), then p(v) < p(u).
73
-1}
2) There exist strictly increasing and continuous real functions f and g such that
f()
= g(O) = 0, and D is the image under (f, g) :R 2 --+ R 2 of the following
union of semicircles:
{(x , y) (E 1R '
(+-
y =
P U
~
P V
)
(U~v
(U,V)EE
A planar depiction (D, p) of T is said to be non-crossing if for all (x, y) E D with
y > 0, a sufficiently small neighborhood of (x, y) in D is homeomorphic to the real
line.
Lemma 3.4.3. In Definition 3.4.2, the property of being a non-crossing planar depiction is independent of the choice of functions f and g, and only depends on p and
T. In other words, any two planar depictions (Di,p) and (D2 ,p) of T are either both
non-crossing or both crossing.
Example 3.4.4. Figure 3-4a shows a particular acyclic directed-graph T = T(V, E)
with IVI = 7, and a choice of depiction function p : V -+ {0, 1, 2, 3, 4, 5, 6} (in
blue). With this choice of p, Figure 3-4b then presents the set D obtained by taking
f(x) = x and g(x) = 1x in Definition 3.4.2. There are five crossings in D, each
marked with a square; these crossings are the points (x, y) E D, y > 0, that are
locally non-homeomorphic to the real line.
Definition 3.4.5. A non-crossing tree is a non-crossing planar depiction of a rooted
tree T = T(V E). Vaguely, T is obtained from an acyclic connected simple graph on
vertex-set V by orienting all of its edges towards a distinguished vertex of T, called
the root of T (e.g. Figure 3-5c).
Remark 3.4.6. In Definition 3.4.5, for one such non-crossing tree (D, p) of T, if r is
the root of T, then necessarily p(r) = 0.
Theorem 3.4.7. Let G = G([n], E) be a simple graph, and consider a spanning
tree T of Gr rooted at r. Then, there exists a unique depiction function p as in
Definition 3.4.2 such that:
i For all edges (i, k) and (j, k) of T, p(i) > p(j) if i < j.
ii Any planar depiction (D,p) of T is a non-crossing tree.
Proof. For any two i, j E [n] with i i j, consider the directed-paths from i and j to
the root r of T. These paths meet initially at a unique vertex rij of T. Let us say
that i -<T j if either 1) rij = i or if. 2) there exist edges (ij, rij) in the path from i to
rij and (ji, rij) in the path from j to rij such that i3 > ji.
Firstly, we verify that the relation rT is a total order on the set [n] of vertices of
G. This is true since for i --<T j and j -<T k with i, j, k E [n]:
74
T =T(V, E):
p:
-6}
D, f (x) = X, g(x) = !X
2
p: V --+ {0,.6}
Five Crossings: m
1
3
2
1
L
S4
-H
2
1
0
4
3
5
6
2.
(b)
(a)
Figure 3-4: Example of a planar depiction, according to Definition 3.4.2.
a. If rij = i and rk=
j, then ri = i.
i = 1'ij
=Tikk
r ir =
r
ri
a.
b. If rij = i and rik
k -A
# j,
then either rik
i or rik
=
=
rik and in the later case
> kj = ki.
trij
rik
r =Jrij
r
k
Sk
b.
c. If rig
iand rgk=j, then ri = rk and ik = ij > ji =k.
rij =
r
k
ik
rjk
--
C.
d. If ri, = i, rik ,4 3 and rij = rik, then ik = ii > ji = jA > k. = ki.
k
=
r
d.
75
rk
e. If rij
7 i, rjk 7
j and rij
<T rjk,
then ik - ij
> ji =
ki.
k
r
ri
_jk
i
e.
f. If rij
$
i, rJk
$j
and rk -<T r
then ik - jk > kj = ki.
k
r'
jk
rTj
f.
Therefore, in all cases, i -< k.
Let f : [n] -+ [n] be the unique linear extension of this chain poset ([n], --<T) and
define p by requiring that p(r) = 0 and p(i) = f(i) for all i E [n]. Clearly then p
satisfies Condition i.
We now want to check that any depiction (D, p) of T is non-crossing. Suppose
on the contrary that one such depiction is crossing. If that is the case, then there
exist edges (j, i) and (m, k) in T such that p(i) < p(k) < p(j) < p(m), and hence
Jm
jk
<kj = iM3 < gm, a contradiction. This proves ii.
To prove that p is the unique bijection [n] U{r} -* {0, 1, 2,
=
... , n} satisfying i-ii, let
us suppose that another depiction function q works as well. Since q is order-reversing,
then for any i, j E [n] with i
$ j
and rij = i, we must have that q(i) < q(j). If instead
rij $ i, j and ij > ji, then Condition i and transitivity imply that q(ij) < q(ji) < q(j),
and then Condition ii shows that q(ij) < q(i) < q(ji) < q(j) since in any planar
depiction of T using q, the depiction of the path from i to rij (or to ij) does not cross
the depiction of the path from j to rij (or to ji). Hence, q(i) < q(j). This shows that
q = p from 1) and 2) above.
Example 3.4.8. Figures 3-5a up to 3-5e offer an example of the unique depiction
function p of Theorem 3.4.7. For the graph G = G([7], E) of Figure 3-5a, we calculate
Gr in Figure 3-5b. We then select a particular spanning tree T of Gr (Figure 3-5c, in
red, left diagram) and root it at r (Figure 3-5c, right diagram). Next, we present an
inductive construction of the depiction function p of Theorem 3.4.7 associated to T.
Figures 3-5d.i-v exhibit an inductive calculation from T of a certain special diagram
D (in red), and the final output of this calculation is fully illustrated in 3-5d.v. This
final diagram 3-5d.v shows a non-crossing tree from which p can be instantly read off
(see table). At every step of the construction, we aim to respect both Conditions i and
76
(a)
(b)
G = G([7], E):
(c)
Gr:
2
5
6
r
1
1
r
4
1
4
5
(d)
i)
ii)
(e)
iv)
iii)
X'=
5
r
r
3 2
r
4
13
r
2
5
47 6 1
4
2
5
1
2
v)
1
di
1
Vertex
r1
-.
.
2l
2
2
7
0
3
5
4
6
3
6
6
4
5
7
3
Xi<C4X(6X7
V
(
a= (1.0.0,1.0.1,1),
) 0.1,23.4.56.7
0.15.2,,4.6,7
07.15,2,3,4.6
067.15,2.3A
067.125,3.4
012567.3.4
0124567.3
01234567
7
-
Figure 3-5: Fully worked out example illustrating the central dogma of Section 3.4.
Theorems 3.4.13 and 3.4.22 are dwelled on in tables 3-5e.i and 3-5e.ii, respectively,
and in particular, fa = p
ii of Theorem 3.4.7, and this is seen to imply the uniqueness of p for this example. In
fact, it is not difficult to observe that the analogous inductive process can be readily
applied to any other example, from which Theorem 3.4.7 follows.
3.4.1
Standard monomials of TG.
Definition 3.4.9. Let G = G([n], E) be a simple graph and let Xa be a standard
monomial of the ideal TG. From a, let us define a bijection fa : {,1,... ,n} -+
[n] U {r} and an r-rooted spanning tree Ta of Gr recursively as follows:
1. The edge-set E(Ta) of Ta will be constructed one edge at a time. Similarly, a
set K will contain at each step the set of values in {0, 1, ... , n} for which fa
has already been defined.
2. Initially, set E(Ta) = 0, fa(O) = r, i = 1, and K = {0}.
Since fa(k) has been defined for all k E K, let us also denote this partiallydefined function by fa (which should not cause any confusion).
Step i.
3. Let (k,
j) be the lexicographically-maximalpair among all pairs such that:
77
a) k c K,
b) j E [n]\fa[K], and
c) for {lo < - - < lm} = fa 1 [NGr(j)
n fa[K]],
we have k = las.
4. From this pair (k, j), set fa(i) = j and E(Ta) = E(Ta) U {(j, fa(k))}.
5. K=KU{i}.
6. i=i+1.
7. Go back to 3 if i < n, otherwise stop.
Proposition 3.4.10. In Definition 3.4.9, both fa and Ta are well-defined. Furthermore, if we set pa = f.1, then pa is the unique function of Theorem 3.4.7 such that
any planar depiction (D,pa) of Ta is a non-crossing tree.
Proof. If the condition of Definition 3.4.9.3.c) can be attained at each step of the
recursion, that is, if for all i E [n] we are able to find at least one such pair of k and j
for which k = 1a, then it is clear that fa is a bijection and Ta (with edge-set E(Ta))
is a spanning r-rooted tree of Gr. It then follows easily that pa is order-reversing.
Now suppose that we are at the i-th step of the recursion, i < n, so that K =
{0,1,... ,i - 1}. Since for 0 $ - = [n]\fa[K] we have that XOutof(G,)+la E TG, then
there must exist at least one j E u- such that a3
outof(G )(j). Therefore, if we write
{lo < ... < lm} = f,-1 [NGr(j) n fa[K]] and observe that in fact m = outof(G)() it
follows that k = la is defined correctly for this choice of j.
Let us now establish the non-crossing condition given the choice of depiction function Pa = fa-1 . Notably, the recursive definition of fa is tailored at making this verification rather simple. Indeed, suppose that there exists a first step of the recursion,
say the i-th step, i < n, where a pair of crossing curves will be formed in any depiction
(D, Pa) of Ta, and let (k, j) be the lexicographically-maximal pair found in this step.
Let also (ko, jo) be the optimal pair found at the io-th step with io < i, such that the
curves representing the edges (jo, fa(ko)) and (j, fa(k)) cross in all pa-depictions of
Ta. Then, ko < k < io < i. This implies that the pair (k, j) is lexicographically-larger
than (ko, jo) and that, during the io-th step, the condition of Definition 3.4.9.3.c) is
also attained for (k, j), so that k = la,. Contradiction.
It remains to prove that pa satisfies Condition i of Theorem 3.4.7, but this follows
immediately from the choice of lexicographically-maximal pairs at each step of the
recursion.
Definition 3.4.11. Let G = G([n], E) be a simple graph, T an r-rooted spanning tree
of Gr, and p the unique depiction function of Theorem 3.4.7 associated to T. Let us
associate with T a vector b(T) E NI") in the following way:
For all i E [n] and unique directed-edge (i, i,) in T, let
b(T)i ={ I j E NGr(i) : PUj) < Aiir) II-
78
Proposition 3.4.12. In Definition 3.4.11, the monomial xb(T) is a standard monomial of the ideal TG.
Proof. Consider the bijective function f : [n] -+ [n] given by f(i) = n + 1 - p(i) for
all i E [n]. Clearly then b(M)-i() 5 OutOf('Gf-1[,)(f-1(i)) < outof(G,,f-1,ill)(r'() + 1,
and we are exactly in the situation of the second part of the proof of Theorem 3.3.6,
so we obtain that xb(T) g
G-
ANG4]
Theorem 3.4.13. Let G = G([n], E) be a simple graph, Xa a standard monomial
of TG, and T an r-rooted spanning tree of Gr with unique depiction function p as
in Theorem 3.4.7. Then, using the notation and functions from Definitions 3.4.93.4.11 and Proposition 3.4.10, we have that b(T.) = a and Tb(T) = T. Hence, the
non-crossing trees obtained from the spanning trees of Gr interpolate in a bijection
between rooted spanning forests of G and standard monomials of TG, in such a way
that every non-crossing tree naturally corresponds to a uniquely determined object
from each of these two sets of combinatorial objects associated to G.
Proof. This is now a straightforward application of the recursive definition of fa (or
of fbcT)). For the first equality, let us suppose that during the i-th step of the recursion
to define fa, so K = {0, 1, ... , i - 1} and i < n, we find a lexicographically-maximal
pair (k, j) with k = la, where {lo < ... < lm} = f-1 [NGr (j) n fa[K]]. Then:
I{f
b(Ta)j
E NGr(j)
Pa(f) < pa(jr)}I
(j ,r)
E NGr U)
fa f() <fa7 1 (fa(k))
E NGr U)
fa 1 (f) <k}I
=
E NGr U)
fa 1(V) <
=
E NGr(j) fa[K] :
=I
=
{
fa(')
< la. 5 i -
1
E E(Ta)}
= aj.
This proves the first equality.
For the second equality, we use induction on N to prove that fb(T) (N) = p-'(N)
for all N E {O, 1, ... , n}, and then to argue that during step N > 1 of the recursion
to define fb(T), N < n, the edge that will be added to the set E (Tb(T)) is an edge of
T. Initially, when N = 0, we have fb(T) (0) = p-1(0) = r and E (Tb(T)) = 0. Suppose
that the result is true for all N < i, i E [n], and let us consider the i-th step of the
recursion, so that K = {0,1,..., i - 1}. By induction, if j E [n]\fb(T)[K] and since
1
fb(T)(k) = p- (k) for all k E K, we have that
{lo
so when
<
...
<lm} = f;l) [NGr(J)
nl fb(T)[K]]
=p [NGr(j) np-1 [K]]
b(T)j < M:
{(ji
lb(T)= lI{eENGr(i):p()<p(jr)}I
lI{tENGr(j)np' [K]:p(e)<p(jr)}I
jr)
E E(T), definition of b(T)}
{ir
E
p-1
[K] }
{definition of l4 and induction}
p(ji)
79
Hence, the choice of lexicographically-maximal pair (k, j) necessarily corresponds to
an edge of T, that is, (j, fb(T)(k)) E E(T). Letting s := p- 1 (i) and (s, sr) E E(T),
that maximal pair selected from T is easily seen to be (p(sr), s), again by the inductive
step and the conditions satisfied by p (and T) from Theorem 3.4.7.
Example 3.4.14. Figure 3-5e.i presents the standard monomial of TG that corresponds to the spanning T tree of Gr in Example 3.4.8. For example, to calculate
(a)4 = a 4 , we find cusp 4 (in black) in Figure 3-5d.v. To the left of cusp 4 in this
diagram, there is exactly one adjacent cusp to 4 through a red arc. This is cusp 5 (in
black), so we say that 5 = 4r. There is exactly one cusp in the diagram strictly to
the left of 5 that is adjacent to 4, that is r. Therefore, a 4 = 1, as in Definition 3.4.11.
Proposition 3.4.15. Let G = G([n], E) be a simple graph. Then, there exists a
bijection between the following sets:
1. The set of acyclic orientations of G.
2. The set of r-rooted spanning trees T of Gr such that if p is the depiction function
for T of Theorem 3.4.7, then for all (i,ir) E E(T) and j E [n] with p(ir) <
E.
p(j) < p(i), we have that {i,j}
Moreover, if T (with depiction function p) corresponds to an acyclic orientation0 of
G under this bijection, then the function f : [n] -+ [n] given by f(m) = n + 1 - p(m)
for all m E [n] is a linear extension of 0, and for (i,ir) E E(T) with i, i E [n], ir
covers i in 0.
Proof. Let us first show that the maximal (by divisibility) standard monomials of TG
are in bijection with the acyclic orientations of G. Let a E Nn be such that Xa 0 TG
but xa+ei E TG for all i E [n]. From the Alexander duality of AG and TG, consider
an acyclic orientation 0 of G such that ai < outdeg(GO)(i) for all i E [n]. Since
ai + 1 > outdeg(G,O)(i) + 1 for all i, then it must be the case that ai = outdeg(G,O) (I
so a = outdeg(G,O). It is well-known and not difficult to prove that the out-degree (or
in-degree) sequences uniquely determine the acyclic orientations of a simple graph,
so this establishes that the maximal standard monomials of TG are in bijection with
the (out-degree sequences of the) acyclic orientations of G.
Now, given an r-rooted spanning tree T of Gr with depiction function p as in
Theorem 3.4.7, let us define an orientation 0 (not necessarily a p.a.o.) of G associated
to T. For all e = {i, j} E E, let:
O(e) =
j)
J(i,
e
p(ir), where (i, ir) E E(T),
if p(j)
otherwise.
Consider the out-degree sequence outdeg(GO) associated to the orientation 0, i.e.
outdeg(G,O)(i) = I{j [n] : (i,j) E O[E]}I for all i E [n]. We then note that b(T), =
outdeg(G,O)(i) for all i, so b(T) = outdeg(,). However, the out-degree sequence
outdeg(GQO) corresponds to an acyclic orientation of G if and only if T satisfies that
80
2
2
0:
6
6
4
1
4
1
r
0
3
6
1
2
2
.
a = (2, 0, 0 2, 3, 1, 2),
xa = x
x x 2
TG
a outdeg(G,0) ................-
r
71
5
7
Figure 3-6: Example of the bijection of Proposition 3.4.15. The selected spanning
tree of Gr (in red) corresponds to the acyclic orientation 0 of G presented.
for all (1, 1r) E E(T) and j E [n] with p(ir) < p(j) < p(i), we have that {i, j}
E
since we require that all edges of E get oriented (or get mapped to directed-edges)
through 0. This proves the main statement.
That f is a linear extension when 0 is an acyclic orientation follows since then,
for (i,j) E O[E], necessarily p(j) < p(ir) < p(i) by the Definition of 0 from T and
p; likewise if (i, r) E E(T) with 1,ir E [n], then ir covers i in 0 since p(ir) > p(j) for
all (i, j) E E(T) and p is order-reversing.
Example 3.4.16. Figure 3-6 illustrates both the statement and proof of Proposition 3.4.15. Firstly, we show an acyclic orientation 0 of a graph G = G([7], E)
(Fig. 3-6, left). Then, we select a particular special spanning tree of Gr (Fig. 3-6, in
red), and calculate the non-crossing tree representation of this spanning tree (Fig. 3-6,
below). Arcs of this lower diagram represent edges of Gr. To each cusp i (in black) of
the diagram with i E [7] = {1, 2, ... , 7}, there is a unique adjacent red arc to the left,
and we let ir (in black) be the other cusp adjacent to the same red arc, e.g. for i = 5
we have 1 = 5 r. Let us orient from right to left every arc of the diagram adjacent to
cusp i if the other cusp adjacent to the arc is either ir or lies to the left of ir, e.g.
the arcs from 5 to 1, 5 to 4, 5 to 3, and 5 to r, get all oriented from right to left.
Doing this for all i, we obtain an orientation of (some of the arcs of) the diagram,
and hence an orientation of Gr. In our example, this orientation yields an acyclic
orientation of Gr, and all edges are assigned an orientation; however, this might not
be the case for several other choices of spanning tree of Gr. Moreover, the restriction
of this acyclic orientation to the edges of G is precisely 0, and this is the bijection of
Proposition 3.4.15.
81
3.4.2
Non-crossing partitions.
Definition 3.4.17. A non-crossing partition of the totally ordered set [0, n] = {0, 1, ...
is a set partition 71 of [0, n] in which every block is non-empty and such that there does
not exist integers i < j < k < 1 and blocks B # B' of 7 with i, k E B and j, 1 E B'.
The set of all non-crossing partitions of [0, n] ordered by refinement
(ref)
forms
a graded lattice of length n, and we will denote this lattice of non-crossing partitions
of [0, n] by NC([o,n]).
Definition 3.4.18. Consider a maximal chain C = {7o -<re 7I -ref ... <ref 7Tn} of
NC([O,n]). For each i E [n], there exists a unique element i E [n] such that i is the
minimal element of its block in Ti_ 1 but i is not the minimal element of its block in
7rt. Let then i = i be the element of the block of i in 7Ti that immediately precedes i.
With this notation, we define a bijection Pc : [n] U {r} -+ [0, n] and an r-rooted
tree TC of (K[nl)r ~'_
K[n]U{r} associated to the chain C in the following way:
pc(r)
=
0,
(PC:)
Pc(i) =i, for all i E [n].
E(Tc)
=
{(p-'(-), p- (i)) : i E [n]}
.
(Tc :)
Proposition 3.4.19. In Definition 3.4.18, both pC and T, are well-defined and moreover, Pc is the function of Theorem 3.4.7 such that any planar depiction (D, Pc) of
Tc is a non-crossing tree.
Proof. That Pc is well-defined is a consequence of the fact that taking the union of
two disjoint blocks in a partition of [0, n] will make exactly one minimal element of
these blocks non-minimal in the newly formed block. Hence, in a maximal chain of
NC([o,n]), every non-zero element of [0, n] stops being minimal in its own block at
exactly one cover relation in the chain, and every cover relation in the chain gives
rise to one such element. That T, is an r-rooted spanning tree of (K[]) r comes from
observing that, since Pc is well-defined, the di-graph Pc o T, on vertex-set [0, n] and
edge-set (i, L) for all i E [n], is a 0-rooted spanning tree of K[o,n]. This is true because
for every i E [n], there exists exactly one edge in Pc o T, of the form (i, j) with j < i,
and these are all the edges of Pc o TC.
To verify that Pc and T, satisfy Condition i of Theorem 3.4.7, suppose on the
contrary that there are edges (i, k), (j, k) E E(Tc) with i <
j
and pc(i) < pc(j). This
means that i was minimal in its block in 7ti 1 but not in 7i, and that both i and
i = pc(k) lied in the same block of 7i. Similarly, j was minimal in yj_1 but not in 7tj,
where it was immediately preceded by j = pc(k) = i. Since j > i, all three i, j and
j belonged to the same block of 7tj, but j = 1 < i = pc(i) < pc(j) = j shows that j
does not immediately precede j in 7Tj, contradiction.
To verify the non-crossing condition, note that if there is a crossing in a depiction
(D, PC) of TC, then there is a smallest i E [n] such that there exists j < i with either
j <.i <J < i or i <j < i < j. In both cases, we observe that {i,i} and {j, a} belong
to different blocks of 7Ti. But then, these two blocks must cross in 7Ti, clearly. This is
a contradiction.
82
,
n}
Definition 3.4.20. Let G = G([n], E) be a simple graph, and let T be an r-rooted
spanning tree of Gr. Suppose that p is the depiction function of Theorem 3.4.7 such
that any depiction (D,p) of T is non-crossing.
From T and p, let us form a chain CT
=
{7To
-ref 71 4ref ...
ref
'n} of partitions
of the set [0, n] in the following way:
1st. Let 7to = {{0}, {1},... , {n - 11, {n}}, and
2nd. for each i E [n], let 7it be obtained from 7i_ 1 by taking the union of the block
that contains p(i) and the block that contains P(ir), where (i, ir) is an edge of
T.
Proposition 3.4.21. In Definition 3.4.20, CT is well-defined and moreover, it is a
maximal chain of partitions in NC(o,nj).
Proof. That CT is a well-defined (maximal) chain of partitions of [0, n] is a consequence of p being a bijection [n] U r -+ [0, n] and of T being a spanning tree of
Gr: We can think of the procedure of Definition 3.4.20 as that of beginning with an
independent set of vertices [n] U r, and then adding one edge of T at a time until we
form T, keeping track at each step of the connected components of the graph so far
formed (and mapping those connected components through p); there are n such steps
and at each step we add a different edge of T. In fact, since T is rooted and p is
order-reversing, if for some i E [n] we consider the edges (1, 1r), . . . , (i, ir) of T that
have been added up to the i-th step in this process (so that the graph in consideration
is a rooted forest), we see that if two numbers k < I in the set [0, n] belong to the
same block B of ni, then either (p'(l'),p-1(k)) is an edge of T for some l' E B with
k < l' < 1 and p-1(l') < i , or there exist k',l' E B with k' < k < l' < 1 such that
(p- 1 (l'),p- 1 (k')) is an edge of T and p'(l') < i.
Suppose now that some of the partitions in CT are crossing, so let us assume that
i is minimal such that 7ri is crossing and let (i, ir) E E(T). Hence, the block Bi in ti
that contains both p(i) and p(ir) crosses with another block BI of 7Tj, so there exist
two consecutive elements i 1 < i 2 of Bi and two consecutive elements Ji < 12 of B,
such that either
a) il < ji < i2 < J2, or
ii <
il < 32 < i2
.
b)
In i_1, il and i 2 belong to different blocks Bil and Bi 2 respectively, and Bi = Bi, LJBi2
(i, 12) and
Moreover, since i was chosen minimally, if a) holds above then B 2
0. As p is
=
B
fl
(jl,j2)
J2)
and
Bi n (jil, j2) = 0, and if b) holds then Bil C (i,
2
order-reversing, so p(ir) < p(i), we see that p(ir) E Bi1 and p(i) E Bj2 , and then that
i1 < p(i) and p(ir) < i 2 . These last two inequalities imply that p(ir)
ii < i2 5
p(i). Also, since p satisfies Condition i of Theorem 3.4.7, we observe that necessarily
Otherwise, as both il and p(ir) belong to the same block Bil of 7ti- 1 and
= P(ir).
Pi
p(ir)
ii, then either:
83
(1) (p-1(l),
r)
is an edge of T for some 1 E Bi, with p(zr) < 1 < ii and p-1 (l) < i,
which cannot hold since ii < p(i), or
(2) there exist k, l c Bil with k < p(ir) < 1 < ii < p(i) such that (p- 1 (l),p-1 (k)) is
an edge of T, which cannot hold because that edge crosses (i, ir) in any depiction
(D, p) of T.
More easily, since i 2 < p(i) and there are no edges of the form (i, 1) in T except for
(z, ir), we must in fact have that i 2 = p(i). It is now clear that if a) or b) holds above
with i1 = p(ir) and i2 = p(i), then in any depiction (D, p) of T we may find an edge
of T that crosses (i, ir), which is impossible.
l
Theorem 3.4.22. Let K[] be the complete graph on [n], T be an r-rooted spanning
tree of (K[n])r
K[n]U{r} with unique depiction function p as in Theorem 3.4.7, and
C = {7T0 -f
7t,i
ef
...
-
7} a maximal chain of NC([O,n]).
r,.
Then, using
the notation and functions of Definitions 3.4.18-3.4.20, we have that T(c) = T and
C(Tc) = C. Hence, the non-crossing trees obtained from the spanning trees of (K[n])r
interpolate in a bijection between rooted spanningforests of K[n] and maximal chains
of the non-crossing partitions lattice NC([o,n]): Every non-crossing tree corresponds
bijectively to an element of each of these two combinatorial sets.
Proof. This is clear from the proofs of Propositions 3.4.19-3.4.21 through the following simple observations.
Firstly, the edges of Tc correspond to the cover relations in C so that an edge (i, ir)
with i e [n] exists in Tc for every minimal element pc(i) in its block of 7i_ 1 that stops
being minimal in its block of 7Ti; the number ir is then recollected by requiring that
Pc(ir) is the immediate predecessor of pc(i) in the newly formed block of 7r. Nextly,
for all i C {n], the i-th cover relation in C(Tc) corresponds to taking the union of the
block that contains pc(i) and Pc(ir). Therefore, C = C(Tc).
Secondly, the i-th cover relation in CT, i c [n], corresponds to taking the union
of the (disjoint) blocks that contain p(i) and P(ir), where (i, ir) is an edge of T.
Moreover, from the second part of the proof of Proposition 3.4.21, p(i) was minimal
in its initial block and p(ir) immediately precedes p(i) in the newly formed block.
But then, the edges of T(cT) are given by all the (i, 4). Hence, T(c = T.
Example 3.4.23. Table 3-5e.ii shows an example of the bijection of Theorem 3.4.22,
presenting the maximal chain of NC([o,7]) corresponding to the spanning tree T of
Gr of Example 3.4.8 (top to bottom of table, blocks separated by commas). Let
us discuss how this list of non-crossing partitions can be calculated from Figure 35d.v. We will inductively define a set of graphs Go, G1,... , G7 , each on vertex-set
[0, 7] = {0, 1, . . , 7} and with edge-sets Eo, E1 , . . , E7 , respectively. Initially, Go has
no edges, so Eo = 0. Suppose then that we have defined Gi- 1 and Ei_ 1 with i < 7,
and that we want to define Gi and Ei. We find cusp i (in black) in Figure 3-5d.v
and note that, to this cusp, there is exactly one red arc adjacent to the left. This
84
.
arc is also adjacent to cusp ir (in black). Let us then read off the blue labellings of
cusps i and ir in Figure 3-5d.v, and say that these are p(i) and p(ir). Then, writing
e := {P(i), P(ir)}, we let Ei = Ei- 1 U {e} and update Gi accordingly. We stop when
G7 is defined. Notably, G7 is a spanning tree. Non-crossing partitions of Table 35e.ii are then, in order, given by the connected components of the spanning forests
Go, G 1 , . . , G7
Corollary 3.4.24 (Germain Kreweras' classic result). The number of maximal chains
in NC([O,n]) is (n + 1)n-1.
Corollary 3.4.25. We have that:
{Bi,...,Bm}ENC([n])
|BiI!, |B2|!,...
,B
!
n)-
(n +
Therefore, using Speicher's exponential formula for NC([n]) [Speicher, 1994} we obtain the classic result:
n=1
Proof. For each {B 1 ,... , Bm} E NC([n]), where b1 < b2 < ... < b, are respectively the minimal elements of B 1, B2 , . . , B, and for each bijection f : [n] -+
[n] such that f is strictly decreasing on each block Bi, i E [m], we can define
an r-rooted spanning tree T of (K[n])r by taking E(T) = {(f(i), r) : i E B1 } U
{(f(i), f(bk - 1)) : i E Bk with k > 1}. If we let p(r) = 0 and p(i) = f-(i) for all
i E [n] above, we can readily check that p is the depiction function of Theorem 3.4.7
associated to T.
Conversely, given an r-rooted spanning tree T with depiction function p as in
Theorem 3.4.7, the partition UE[n] {p(i) E [n] : (i, k) E T} is an element of NC([n]).
Hence, since given a partition {B 1 ,..., B,} E NC([n]), there are ()B 1I!,IB2 n . IBmJ!)
choices for f above, the result follows.
3.5
3.5.1
Applications.
Random Acyclic Orientations of a Simple Graph: Markov
Chains.
Definition 3.5.1. Let G = G(V, E) be a connected (finite) simple graph. A simple
random walk on G is a Markov chain (Vt)t=0,1,2,..., obtained by selecting an initial
vertex vo E V, and then for all t > 1, selecting vt E V from a uniform distribution
on the set NG(vt-1). If P is the Markov transition matrix for a simple random walk
on G, then for u,v E V:
G(u)
="
0
85
if v G NG(u),
otherwise.
Theorem 3.5.2. The Markov chain of Definition 3.5.1 is always irreducible. Furthermore, it is aperiodic if and only if G is not bipartite.
If for all w G V, we let 7T, :
G(),
21E1' then for any pair of vertices u,v E V, we
have that:
7
Consequently, since EV
=
7
Tupuv.
pEvu =
1, random walks on G are reversible and they have a
unique stationary distribution given by
= (TrV)VEV,
7
so that:
N
7Tv
= lim.
N-+oo N
Pr[vt =-v], for all v E V.
1
(3.5.1)
Moreover, if G is not bipartite, then:
7V
= lim Pr[vt
t-+00
=
v], for all v c V.
(3.5.2)
Definition 3.5.3 (Card-Shuffling Markov Chain, see also Athanasiadis and Diaconis
{2010}). Let G = G(V, E) be a simple graph with|V| = n > 3, and select an arbitrary
bijecttve-map fo : V -+ [n] (regarded as a labelling of V). Let us consider a sequence
(ft)t=o,1,2,... of bijective maps V -+ [n] such that for t > 1, ft is obtained from ft-1
through the following random process: Let vt E V be chosen uniformly at random,
and let,
n
if v = vt,
ft (v)=
ft i(v) - 1
if ft i(v) > ft i(v),
ft-i(v)
otherwise.
Consider now the sequence of acyclic orientations (Ot)t=o.1,2,... of G induced by the
labellings (ft)t=0,1,2,..., so that for all e = {u, v} E E and t > 0, we have that Ot(e) =
(u,v) if and only if ft(u) < ft(v). The sequence (Ot)t=o,1,2... is called the Cardshuffling (CS) Markov chain on the set of acyclic orientations of G.
Equivalently, we can define this Markov chain by selecting an arbitrary acyclic
orientation Oo of G, and then for each t > 1, letting Ot be obtained from Ot_1 by
selecting vt e V uniformly at random and taking, for all e c E:
Ot(e) =
Ot-i(e)
(v,vt)
if vt g e,
if e= {v,vt}.
Theorem 3.5.4. The Card-Shuffling Markov chain of G in Definition 3.5.3 is a welldefined, irreducible and aperiodic Markov chain on state space equal to the set of all
acyclic orientations of G; its unique stationary distribution 7rC is given by:
S=
,e(O)
for all acyclic orientations 0 of G,
where e(O) denotes the number of linear extensions of the induced poset (V, <o).
Proof. If we consider instead the Markov chain (ft)t=o,1,2,..., whose set of states is
the set of all bijections V -+ [n], it is not difficult to observe that this Markov chain
86
is irreducible and aperiodic (see below), and hence that it has a unique stationary
distribution n satisfying Equations 3.5.2. By the symmetry of the set of all bijective
labelings V -+ [n], or simply by direct inspection of the stationary equations for this
Markov chain (since every state can be accessed in one step from exactly n different
states and each one of these transitions occurs with probability -), we obtain that
for all bijective maps f : V -+ [n]. Hence, by the aforementioned construction
7T =
of the Card-Shuffling (CS) Markov chain of G from bijective labellings of V, we must
have that this CS chain is also irreducible (since each labelling is accessible from every
other labelling, hence each acyclic orientation from every other acyclic orientation),
aperiodic (since both Pr[ft = ft-I|ft-1] > 0 and Pr[Ot = Ot-1|Ot-1] > 0 for
all t > 1), and has a unique stationary distribution 7rCs, necessarily then given by
for every acyclic orientation 0 of G, from Equations 3.5.1.
i(!
710 =
Definition 3.5.5 (Edge-Label-Reversal Stochastic Process). Let G = G(V, E) be a
connected simple graph with |VI = n, and select an arbitrary bijective map fo : V -+
[n] (regarded as a labelling of V). Let us consider a sequence (ft)t=0,1,2,... of bijective
maps V -+ [n] such that for t > 1, ft is obtained from ft-1 through the following
random process: Let et = {ut, vt} E E be chosen uniformly at random from this set,
and let,
if V = vt,
ifv=ut,
ft-i(ut)
ft_1(Vt)
ft_1(v)
ft(v)=
otherwise.
Consider now the sequence of acyclic orientations (0t)t=0,1,2,... of G induced by the
labellings (ft)t=0,1,2,..., so that for all e = {u, v} E E and t > 0, we have that Ot(e) =
(u, v) if and only if ft(u) < ft(v). The sequence (Ot)t=,1,2,... is called the Edge-LabelReversal (ELR) stochastic process on the set of acyclic orientations of G.
Theorem 3.5.6. The Edge-Label-Reversal stochastic process of G in Definition 3.5.5
satisfies that, for every acyclic orientation 0 of G:
N
(irELR)
=ELR
:.
Jim
Lt
_L
=0]-
e(Q)
t=1
where e(O) denotes the number of linear extensions of the induced poset (V,
this result holds independently of the initial choice of 00.
O), and
Proof. Consider the simple graph H on vertex-set equal to the set of all bijective
maps V -+ [n], and where two maps f and g are connected by an edge if and only if
there exists {u, v} E E such that f(u) = g(v), f(v) = g(u), and f(w) = g(w) for all
w E V\{u, v}. Since G is connected, a standard result in the algebraic theory of the
symmetric group shows that H is connected, e.g. consider a spanning tree T of G;
then, any permutation in 6v can be written as a product of transpositions of the form
(u v) with {u, v} E E(T). Moreover, by considering the parity of permutations in
6V, we observe that H is bipartite. Now, the sequence (ft)t=,1,2,... of Definition 3.5.5
87
is precisely a simple random walk on H, and the degree of each bijective map f :
V -* [n] in H is clearly JE|, so the stationary distribution for this Markov chain is
uniform. Necessarily then, the result follows from the construction of (Ot)t=o,1,2,... and
Equations 3.5.1.
Definition 3.5.7 (Sliding-(n + 1) Stochastic Process). Let G = G(V, E) be a connected simple graph with|V| = n, and consider the graphGr. Let us select an arbitrary
bijective map fo : V LI {r} -+ [n + 1], which we regard as a labelling of the vertices
of Gr, and define a sequence (ft)t=o,1,2... of bijective maps V Li {r} -* [n + 1] such
that for t > 1, ft is obtained from ft-1 through the following random process: Let
Vt_1 E V U {r} be such that ft_1(vt-_) = n + 1, and select vt G NGr(vt-1) uniformly
at random from this set. Then,
ft(v)=
n n+1
ifv=vt,
ft-1(vt)
ft_1(v)
if V t=v_,
otherwise.
Consider now the sequence of acyclic orientations (Ot)t=0,1,2,... of Gr induced by the
labellings (ft)t=0,1,2,.., so that for all e = {u, v} E E(Gr) and t > 0, we have that
Ot(e) = (u, v) if and only if ft(u) < ft(v). The sequence (Ot)t=0,1,2,... is called the
Sliding-(n + 1) (SL) stochastic process on the set of acyclic orientations of Gr.
Theorem 3.5.8. The Sliding-(n + 1) stochastic process of Gr of Definition 3.5.7
satisfies that, if Sr is the set of all acyclic orientations of Gr whose unique maximal
element is r, then Z0 Pr[Ot
Sr3= oc and for every 0 E Sr:
(nSL )O = 7TSL:0
liM
N-+o
LN
Nt=
Nt=
Pr[Ot =0]
Pr[ Ot
Sr ]
e(O1)
n!
where Ov is the restriction of 0 to E (hence an acyclic orientation of G) and e(Ov)
denotes the number of linear extensions of the induced poset (V, oi ). These results
hold independently of the initial choice of 00.
Proof. Consider the simple graph H on vertex-set equal to the set of all bijective
maps V U {r} -+ [n + 1], and where two maps f and g are connected by an edge if
and only if there exists {u, v} E E(Gr) such that f(u) = g(v) = n + 1, f(v) - g(u),
and f(w) = g(w) for all w E V\{u, v}. If two bijective maps
f, g
: V U {r}
-+
[n + 1]
differ only in one edge of Gr, so that f(u) = g(v) 7 n + 1 and f(v) = g(u) $ n + 1 for
some {u,.v} c E(Gr), but f(w) = g(w) for all w E V\{u, v}, then we can easily but
somewhat tediously show that f and g belong to the same connected component of
H, making use of the facts that vertex r is adjacent to all other vertices of Gr and that
G is connected. But then, the proof of Theorem 3.5.6 shows that H is a connected
graph. Now, the sequence (ft)t=o,1,2,... of Definition 3.5.7 is a simple random walk
on H, and the degree of a bijective map f : V -+ [n] in H is clearly dc,(vi), where
vf c V depends on f and is such that f(vf) = n + 1, so the stationary distribution
88
7r for this Markov chain satisfies that 7Tf = c - dG (vf), for some fixed normalization
constant c E R+. The vertices of H that induce acyclic orientations of Gr from the
set Sr are exactly the bijective maps f : V Li {r} -+ [n + 1] such that f(r) = n + 1,
and for these we have that 7Tf = c - n. The result then follows from the construction
of (Ot)t=0,1,2,... and from Equations 3.5.1.
Cov(0t_ 1 ) :=
{e E Ot-i[E] : e represents a cover relation in (V,
os)}
,
Definition 3.5.9 (Cover-Reversal Random Walk). Let G = G(V, E) be a simple
graph with |Vj = n, and select an arbitrary acyclic orientation Oo of.G. Let us
consider a sequence (Ot)t=0,1,2,... of acyclic orientations of G such that for t > 1, Ot
is obtained from Ot_1 through the following random process: Let (u, v) be selected
uniformly at random from the set,
and for all e E E, let,
if e = {u, v},
(v,u)
Ot(e) =
Oti(e) otherwzise.
The sequence (Ot)t=,1,2,... is called the Cover-Reversal (CR) random walk on the set
of acyclic orientations of G.
Theorem 3.5.10. The Cover-Reversal random walk in G of Definition 3.5.9 is a
simple 2-period random walk on the 1-skeleton of the clean graphical zonotope ZG of
Theorem 3.2.15 (hence, on a particularsimple connected bipartite graph on vertex-set
equal to the set of all acyclic orientations of G), and its stationary distribution 7rcR
satisfies that, for every acyclic orientation 0 of G:
7r
R =
c - ICOv(0),
where c E R+ is a normalization constant independent of 0.
Proof. From the proof of Theorem 3.2.15, the edges 'of ZG are in bijection with the
set of all p.a.o.'s 0 of G such that if 2 is the connected partition associated to 0, then
II = n - 1. Hence, the edges of ZG are in bijection with the set of all pairs of the
form (e, 0), where e E E and 0 is an acyclic orientation of the graph G/e, obtained
from G by contraction of the edge e. The two vertices of ZG adjacent to an edge
corresponding to a (e, 0) with e = {u, v} are, respectively, obtained from the acyclic
orientations 01 and 02 of G such that 0 1 (e) = (u, v), 0 2 (e) = (v, u), and such that
021E\, are naturally induced by 0 (e.g. see Definition 3.2.4). Necessarily
then, both (u, v) and (v, u) correspond respectively to cover relations in the posets
(V, o,) and (V, 02), since otherwise the orientation 0 of G/e would not be acyclic.
On the other hand, given an acyclic orientation 01 of G and an edge (u, v) E 01 [E]
such that v covers u in (V, Oi), then, reversing the orientation of (only) that edge
in 01 yields a new acyclic orientation 02 of G, so (v, u) E 02 [E]. Otherwise, using a
directed-cycle formed by edges from 02 [E], which must then include the edge (v, u),
011E\e =
89
we observe that the relation u <O, v is a consequence of other order relations in
(V, :O5) and v does not cover u there. This is a contradiction, and it furthermore
implies that both 01 and 02 naturally induce a well-defined acyclic orientation 0 of
G/{u, v}.
Hence, the Cover-Reversal random walk of G corresponds to a simple random walk
on the 1-skeleton of ZG (or of Zrentra ) and the result follows now from Theorem 3.5.2,
since this graph is connected and bipartite, clearly.
Remark 3.5.11. Variants of the Cover-Reversal random walk on G, obtained for
example by flipping biased coins at each step, can be used to obtain stochastic processes that converge to a uniform distribution on the set of acyclic orientations of G.
However, these variants are clearly not very illuminating or efficient.
Definition 3.5.12 (Interval-Reversal Random Walk). Let G = G(V, E) be a simple
graph with |VI = n, and select an arbitrary acyclic orientation 00 of G. Let us
consider a sequence (Ot)t=0,1,2,... of acyclic orientations of G such that for t > 1, Ot is
obtained from Ot_1 through the following random process: Let {u, v} c E be selected
uniformly at random from this set, with (u, v) E Ot_1[E], and for all e = {x, y} c E
with (x, y) G Ot 1 [E], let,
(e)
OtOt
e) -
{
,'X)
(xY)
=
Ot- (e)
if u o, X <Oil Y
otherwise.
Oii V,
The sequence (Ot)t=0,1,2,... is called the Interval-Reversal (IR) random walk on the set
of acyclic orientations of G.
Lemma 3.5.13. Let G = G(V, E) be a szmple graph and let 0 be any given acyclic
orientation of G. For an arbitrary edge {u, v} e E, say with (u, v) E 0 [E], let us
define a new orientation O{,,) of G by requiring that, for all e = {x, y} E E with
(x, y) E 0[E]:
1,, (e)
if u <0 x <o y
(y, x)
(x, y) = O(e)
o v,
otherwise.
.
Then, Oul is also an acyclic orientation of G and, furthermore, (OsUe), = 0.
Additionally, for any choice of e 1 , e 2 E E, we have that e, = 0e2 if and only if
el = e 2
Proof. Suppose on the contrary that 01,, is not an acyclic orientation of G. Then,
there exists at least one directed-cycle C C Ou [E] that has the following form:
For E
=
{{x,y}
E: u <o x <oy <o v},
there exists k E P and pairwise disjoint non-empty sets,
k
Pl, Q1, P2,Q 2 , .,Pk,Qk
O ,,[E with C =
90
i=1
(Pi U Qi),
such that for all i E
[k,
Pi = {(p _1,p3Wj)}
OI , [EJ
-i={(1,
pi,,
],
Cj)Osua,[E]\ (Ogu,,g [Eo, .]),
qO and qIQ 1= po
,
:= Po.
where p
This is true simply because any directed-cycle in Ou
edges from both EoV) and E\Egv.
Since
(1) 0 and Ov
(2) u <o p
1
[E] must necessarily involve
agree on E\EV)
7,p2
o v, and
(3) both q1 = pl I and q1
=
then:
uo
q1o
q <o...
q50 q 1
1
o0 v.
In particular, by definition, {qj, q} EEv) . This contradicts the construction of C,
hence Ou,,4 is also an acyclic orientation of G.
= 0, it suffices to check that if for some {x, y} E E with
)
To prove that (O,
u, then in fact u <o x <o
y <ouv x <o
[E] we have that v <o
(y, x) E Ov
y <o v. Somewhat analogously with the previous argument, suppose on the contrary
that there exists some {x, y} E E with (y, x) E Ov [E] for which the condition fails
O{ 1,, [E] such that
to hold. Then, inside any directed-path P = {(pj-, pj)}j=i,...,1,
containment) sub(by
a
maximal
exist
(y, x) E P, po = v, and pipi = u, there must
path Q = {(qj-, qj)} j=,...,IQI C P such that (y, x) E Q C O{Ugg[E]\ (O{g [EoV)]).
o qiQI o v, and
Necessarily then, u <o qO <o qIQI O v, so u 0 qO 0 y <0 x
hence {y, x} E E ). This is a contradiction.
The last statement is a simple consequence of observing that, for every choice of
{u, v} E E, u and v determine a unique interval inside each of the posets (V, :o),
where 0 is an acyclic orientation of G: A non-empty closed interval of a finite poset
is uniquely determined by its maximal and minimal elements.
D
Proposition 3.5.14. In Lemma 3.5.13, consider the simple graph AOnt*r on vertexset equal to the set of all acyclic orientations of G, and in which two acyclic orientations 01 and 02 of G are connected by an edge, if and only if there exists {u, v} E E
= 02. Then, AONte* is an |E|-regularconnected graph.
such that (0j1)
Proof. Firstly, let us note that AOiner is indeed a well-defined simple graph (so it
does not have loops or multiple edges) per the three main statements of Lemma 3.5.13.
Now, we point out that A QONte contains as a spanning sub-graph the 1-skeleton of the
(clean) graphical zonotope ZG since, colloquially, all cover-reversals are also intervalreversals. Hence, since the later graph has been observed to be connected in the proof
91
i~
(a)
(b)
Figure 3-7: Examples of Definitions 3.5.9 and 3.5.12 for the 4-cycle C4. In 3-7a,
we present the 1-skeleton of the graphical zonotope of C4, a rhombic dodecahedron,
where the Cover-Reversal random walk runs; notably, it is not a regular graph. If
four diagonals are added to the graph as shown in 3-7b, we obtain a 4-regular graph,
A OPt in Proposition 3.5.14, where the Interval-Reversal random walk runs.
of Theorem 3.5.10, then AO
have degree JEl, clearly.
te
is also connected. Every vertex of this graph must
Theorem 3.5.15. The Interval-Reversal random walk in G of Definition 3.5.12 is a
simple random walk on the graph AO ter of Proposition3.5.14 (hence, on a particular
regular connected graph on vertex-set equal to the set of all acyclic orientationsof G),
and its stationary distributionn satisfies that, for every acyclic orientation0 of G:
1
IR
S-IXG~(-I)'
where IXG(-1)l is the number of acyclic orientations of G (Stanley, 1973].
Proof. That the Interval-Reversal random walk of G corresponds to a simple random
walk on A 0
'ter
is a direct consequence of Lemma 3.5.13. That A Ointe is connected
and JEJ-regular is the content of Proposition 3.5.14, so we can now rely on Theorem 3.5.2 to obtain the result.
3.5.2
Acyclic Orientations of a Random Graph.
This short subsection is aimed at proving a surprising formula for the expected number
of acyclic orientations of an Erd6s-R6nyi random graph from G , with p E (0,1).
This formula will follow from the results of Section 3.4, and more specifically from
those of Subsection 3.4.1.
92
Definition 3.5.16. Let n E P. A parking function of [n] is a vector a E Nin] such
a( 2 ) < .. - < a,(n), we have that a,(j) 5 i - 1 for
that for any U E EIn] with a,(,2)
all i E [n]. The set of all parking functions of [n] will be denoted by Park[n].
{i E [n]
For a E NH"], let us write Area(a) := a, + a2 + - + an and supp(a)
ai > 0}.
G[n),p. Write q := 1 - p. If
Theorem 3.5.17. Let n E P, p C (0, 1), and G
we let K[n] denote, as usual,
of
G,
and
orientations
IXG(-1)I is the number of acyclic
the complete graph on vertex-set [n], then we have:
E[IxG(-1
)
isupp(a).
Area(a)
aE[n
(353)
Proof. We make use of Proposition 3.4.15. In general, for any simple graph H on
vertex-set [n] (as G here and the complete graph K[n]), we will let TrH be the set
of all r-rooted spanning trees of Hr. Now, for T E TrK[n], we will say that T is
useful if T E TrG and its unique depiction function p of Theorem 3.4.7, satisfies the
conditions of Proposition 3.4.15. Then:
Pr[T is useful]
E
E[IXG(-1)j)
TETrKn]
(Pr[{i,j}
Pr[T E TrG])
E(Gr) for all ij
E [n],
TETrKnl
(i, ir) E E(T), p(ir) < P(j) < P(i)]
J
p() TCETrK[n
Pr[{i,j} V E(Gr) for all j E [n], P(ir) <
iE [n]
(i0r)CEkT)
p(j) < p(i)]
i
n
~pTr
=
iE[n]
(i,ir)EE(T)
TETrKn]
( )
= (n)
qP(i> 41P(ir)
>
TETrK[nf
()Z
q
( )
(n)
aEPark
E[n]1P(ir)
Area(a)
q
as we wanted.
93
n]a> }
Ii
pl{iElnj:ai>O}l
k-Neighbor Bootstrap Percolation.
3.5.3
Definition 3.5.18. Let G = G([n], E) be a finite simple graph, k G P, and A C
[n]. The k-neighbor bootstrap percolation on G with initial set A, is the process
{At~t-0,1,2.., where A 0 = A and At = At_ 1 U {i E fn] : ING(i) n At-1|
k} for all
t > 1. The closure of A is the set cl(A) := U>oA, and we say that A percolates in
G if cl(A)
=
[n].
Question 3.5.19. Given a graph G as in Definition 3.5.18, what is the minimal size
JAl of a set A C [n) such that A percolates in G?
Definition 3.5.20. For fixed G and k as in Definition 3.5.18, let C(G,k) := { C n]
outof(G,,)(i) < k for all z E a}. The k-bootstrap percolation ideal Bc(G,k) of G is the
square-free monomial ideal of k[x1,... , x,,] generated as:
Bc(G,k)
K
H7Ji :
u
E C(Gk)).
iEU
Proposition 3.5.21. In Definitions 3.5.18-3.5.20, the function that associates a
standard monomial xb
Bc(G,k), b e N " , with the set of vertices {i E [n] : bi = 0}
of G, restricts to a bijection between the set of all square-free standard monomials of
Bc(G,k) and the set of all A C [n] such that A percolates in G.
Collcquialy, the percolating sets of G are in bijection with the (complements of)
supporting sets of standard monomials of the ideal Bc(G,k)Proof. Let A
I[n] be such that cl(A) C [n], and consider the set U := [n]\cl(A).
Necessarily, every element of a must have fewer than k neighbors inside cl(A), so
outof(Ga)(i) < k, for all i c a. This implies that U E C(G,k), and x' := J
xC
Bc(G,k)Bc(G,k)
But then, since a C A := [n]\A, we have that x'Ix A := laXzi so
xA
E
as well.
On the contrary, if xA E Bc(G,k) for some A C [n], there must exist some U E C(G,k)
such that x'lxA Necessarily then, cl(A) C [n]\u, since it is never possible to percolate
the elements of a during a k-bootstrap percolation on G from an initial set disjoint
from u, as A here.
LI
94
95
Bibliography
David Aldous and Jim Fill. Reversible markov chains and random walks on graphs,
2002.
David J Aldous. The random walk construction of uniform spanning trees and uniform
labelled trees. SIAM Journal on Discrete Mathematics, 3(4):450-465, 1990.
Drew Armstrong, Brendon Rhoades, and Nathan Williams. Rational associahedra
and noncrossing partitions. arXiv preprint arXiv:1305.7286, 2013.
Christos A Athanasiadis and Persi Diaconis. Functions of random walks on hyperplane
arrangements. Advances in Applied Mathematics, 45(3):410-437, 2010.
J6zsef Balogh, B6la Bollobds, and Robert Morris. Bootstrap percolation in three
dimensions. The Annals of Probability, pages 1329-1380, 2009.
Dave Bayer and Bernd Sturmfels. Cellular resolutions of monomial modules. In J.
reine angew. Math. Citeseer, 1998.
Matthias Beck and Sinai Robins. Computing the continuous discretely: Integer-point
enumeration in polyhedra. Springer, 2007.
B61a Bollobts. The chromatic number of random graphs. Combinatorica, 8(1):49-55,
1988.
Carl Wilhelm Borchardt. Ueber eine der interpolation entsprechende darstellung der
eliminations-resultante. Journal fur die reine und angewandte Mathematik, 57:
111--121, 1860.
Graham Brightwell and Peter Winkler.
225-242, 1991.
Counting linear extensions.
Order, 8(3):
Andrei Broder. Generating random spanning trees. In Foundations of Computer
Science, 1989., 30th Annual Symposium on, pages 442-447. IEEE, 1989.
Andries E Brouwer and Willem H Haemers. Spectra of graphs. Springer, 2011.
Denis Chebikin and Pavlo Pylyavskyy. A family of bijections between g-parking
functions and spanning trees. Journal of Combinatorial Theory, Series A, 110(1):
31-41, 2005.
96
Fan RK Chung. Spectral graph theory, volume 92. American Mathematical Soc., 1997.
Henry H Crapo and Gian-Carlo Rota. On foundations of combinatorial theory. 2.
combinatorial geometries. Studies in Applied Mathematics, 49(2):109, 1970.
Imre Csiszdr, Janos Koerner, Laszlo Lovasz, Katalin Marton, and Gabor Simonyi.
Entropy splitting for antiblocking corners and perfect graphs. Combinatorica, 10
(1):27-40, 1990.
Satyan L Devadoss. A realization of graph associahedra. Discrete Mathematics, 309
(1):271-276, 2009.
Anton Dochtermann and Raman Sanyal. Laplacian ideals, arrangements, and resolutions. Journal of Algebraic Combinatorics, pages 1-18, 2012.
Paul Edelman, Takayuki Hibi, and Richard P Stanley. A recurrence for linear exten-
sions. Order, 6(1):15-18, 1989.
Miroslav Fiedler. A property of eigenvectors of nonnegative symmetric matrices and
its application to graph theory. Czechoslovak Mathematical Journal, 25(4):619-633,
1975.
Miroslav Fiedler. Matrices and graphs in geometry, volume 139. Cambridge University
Press, 2011.
Alex Fink and Benjamin Iriarte G. Bijections between noncrossing and nonnesting
partitions for classical reflection groups. Portugalice Mathematica, 67(3):369-401,
2010.
Lester Randolph Ford Jr and Delbert Ray Fulkerson. Flows in networks. Princeton
Landmarks in Mathematics and Physics. Princeton university press, 2010.
Tibor Gallai. Transitiv orientierbare graphen. Acta Mathematica Hungarica, 18(1):
25-66, 1967.
Geoffrey R Grimmett and Colin JH McDiarmid. On colouring random graphs. In
MathematicalProceedings of the Cambridge PhilosophicalSociety, volume 77, pages
313-324. Cambridge Univ Press, 1975.
Benjamin Iriarte G.
Graph orientations and linear extensions.
arXiv preprint
arXiv:1405.4880, 2014.
Jeff Kahn and Jeong Han Kim.
Entropy and sorting.
Journal of Computer and
System Sciences, 51(3):390-399, 1995.
Jonathan A Kelner and Aleksander Madry. Faster generation of random spanning
trees. In Foundations of Computer Science, 2009. FOCS'09. 50th Annual IEEE
Symposium on, pages 13-21. IEEE, 2009.
97
Caroline J Klivans and Ed Swartz. Projection volumes of hyperplane arrangements.
Discrete & Computational Geometry, 46(3):417-426, 2011.
Ldszl6 Lovisz. Random walks on graphs: A survey. Combinatorics, Paul erdos is
eighty, 2(1):1-46, 1993.
Ldszl6 Lovisz and Peter Winkler. Efficient stopping rules for markov chains. In
Proceedings of the twenty-seventh annual A CM symposium on Theory of computing,
pages 76-82. ACM, 1995a.
Liszl6 Lovisz and Peter Winkler. Exact mixing in an unknown markov chain. the
electronic journal of combinatorics, 2(R15):2, 1995b.
Madhusudan Manjunath and John Wilmes. Minimal free resolutions of the g-parking
function ideal and the toppling ideal. arXiv preprint arXiv:1210.7569, 2012.
Colin McDiarmid. On the chromatic number of random graphs. Random Structures
& Algorithms, 1(4):435-442, 1990.
Russell Merris. Laplacian graph eigenvectors. Linear algebra and its applications,278
(1):221-236, 1998.
Ezra Miller. Alexander duality for monomial ideals and their resolutions.
preprint math/9812095, 1998.
arXiv
Ezra Miller and Bernd Sturmfels. Combinatorialcommutative algebra, volume 227.
Springer, 2005.
Fatemeh Mohammadi and Farbod Shokrieh. Divisors on graphs, connected flags, and
syzygies. InternationalMathematics Research Notices, page rnt186, 2013.
Rolf H Mbhring. Almost all comparability graphs are upo. Discrete mathematics, 50:
63-70, 1984.
Alexander Postnikov. Permutohedra, associahedra, and beyond. InternationalMathematics Research Notices, 2009(6):1026-1106, 2009.
Alexander Postnikov and Boris Shapiro. Trees, parking functions, syzygies, and deformations of monomial ideals. Transactions of the American Mathematical Society,
356(8):3109-3142, 2004.
Jorge L Ramirez-Alfonsin and Bruce A Reed. Perfect graphs. Series in Discrete
Mathematics and Optimization. Wiley-Interscience, 2001.
CM Reidys. Acyclic orientations of random graphs. Advances in Applied Mathematics,
21(2):181-192, 1998.
Kyoji Saito. Principal 7-cone for a tree. Advances in Mathematics, 212(2):645-668,
2007.
98
Raman Sanyal. Constructions and obstructions for extremal polytopes. PhD thesis,
PhD thesis, Technische Universitdt Berlin, 2008.
Carla D Savage and Cun-Quan Zhang. The connectivity of acyclic orientation graphs.
Discrete mathematics, 184(1):281-287, 1998.
Roland Speicher.
Multiplicative functions on the lattice of non-crossing partitions
and free convolution. Mathematische Annalen, 298(1):611-628, 1994.
Daniel Spielman. Spectral graph theory. Lecture Notes, Yale University, pages 740-
0776, 2009.
Grzegorz Stachowiak. The number of linear extensions of bipartite graphs. Order, 5
(3):257-259, 1988.
Richard P Stanley. Acyclic orientations of graphs. Discrete Mathematics, 5(2):171-
178, 1973.
Richard P Stanley. Two poset polytopes. Discrete & Computational Geometry, 1(1):
9-23, 1986.
Richard P Stanley. Parking functions and noncrossing partitions. Electron. J. Combin,
4(2):R20, 1997.
Richard P Stanley. Hyperplane arrangements, parking functions and tree inversions.
In Mathematical Essays in Honor of Gian-Carlo Rota, pages 359-375. Springer,
1998.
Richard P Stanley. Enumerative Combinatorics, Vol. 2:. Cambridge Studies in Advanced Mathematics. Cambridge University Press, 2001.
Richard P Stanley. An introduction to hyperplane arrangements. In Lecture notes,
IAS/Park City Mathematics Institute. Citeseer, 2004.
Richard P Stanley. Enumerative Combinatorics, Vol. 1:. Cambridge Studies in Advanced Mathematics. Cambridge University Press, 2011.
JJ Sylvester. On the change of systems of independent variables. Quarterly Journal
of Mathematics, 1:42-56, 1857.
99