EDGE-CHOOSABILITY OF CUBIC GRAPHS AND
THE POLYNOMIAL METHOD
by
Andrea Marie Spencer
B.MATH., University of Waterloo, 2008
a Thesis submitted in partial fulfillment
of the requirements for the degree of
Master of Science
in the Department
of
Mathematics
c Andrea Marie Spencer 2010
SIMON FRASER UNIVERSITY
Spring 2010
All rights reserved. However, in accordance with the Copyright Act
of Canada, this work may be reproduced, without authorization, under
the conditions for Fair Dealing. Therefore, limited reproduction of this
work for the purposes of private study, research, criticism, review and
news reporting is likely to be in accordance with the law,
particularly if cited appropriately.
APPROVAL
Name:
Andrea Marie Spencer
Degree:
Master of Science
Title of Thesis:
Edge-Choosability of Cubic Graphs and
the Polynomial Method
Examining Committee:
Dr. John Stockie,
Professor of Mathematics
Chair
Dr. Luis Goddyn,
Senior Supervisor,
Professor of Mathematics
Dr. Ladislav Stacho,
Supervisor,
Professor of Mathematics
Dr. Matthew DeVos,
Internal/External Examiner,
Professor of Mathematics
Date Approved:
April 7, 2010
ii
Abstract
A graph is k-edge-choosable if for any assignment of a list of at least k colours to each edge,
there is a proper edge-colouring of the graph such that each edge is assigned a colour from its
list. Any loopless cubic graph G is known to be 4-edge-choosable by an extension of Brooks’
Theorem. In this thesis, we give an alternative proof by relating edge-choosability to the
coefficients of a certain polynomial using Alon and Tarsi’s Combinatorial Nullstellensatz.
We interpret these coefficients combinatorially to show that the required edge-colourings
exist. Moreover, we show that if G is planar with c cut edges, then all but 3c of the edges
of G can be assigned lists of at most 3 colours.
iii
“He gives wisdom to the wise
and knowledge to those who understand.”
— Daniel 2:21
iv
Acknowledgments
I would like to thank Dr. Luis Goddyn, my senior supervisor, for his support, guidance and
patience, as well as for suggesting this problem. I would also like to thank Dr. Ladislav
Stacho, my co-supervisor, for his support and especially for his help in bringing me to Simon
Fraser University. I am grateful to all the members of my examining committee for carefully
reading this thesis and sharing their insights. Thanks also to all my friends and colleagues
at SFU for making my time here both enjoyable and enlightening.
My abundant gratitude goes to my family for their constant support, love, and prayers.
And my thanks go to Kael, not least of all for keeping me sane.
I would like to acknowledge the contributions made by Dr. Goddyn and Dr. Sabin Cautis
(now of Columbia University), who began work on this problem during an undergraduate
research project of the latter.
I acknowledge the financial support of the Natural Sciences and Engineering Research
Council of Canada (NSERC) and of Simon Fraser University. I also acknowledge the
IRMACS Centre for hosting me during my studies.
v
Contents
Approval
ii
Abstract
iii
Quotation
iv
Acknowledgments
v
Contents
vi
List of Figures
viii
1 Introduction
1
2 The Polynomial Method and Star Labellings
4
3 Cubic Graphs
8
3.1
3.2
2-Edge-Connected Cubic Graphs . . . . . . . . . . . . . . . . . . . . . . . . .
8
3.1.1
Hamiltonian 2-Factor . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
3.1.2
2-Factor Containing Multiple Cycles . . . . . . . . . . . . . . . . . . . 10
General Cubic Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.2.1
Structure of the Graph . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.2.2
Edge Weighting of the Graph . . . . . . . . . . . . . . . . . . . . . . . 17
3.2.3
Star Labellings of the Graph . . . . . . . . . . . . . . . . . . . . . . . 21
3.2.4
Proof of Theorem 1.0.3 . . . . . . . . . . . . . . . . . . . . . . . . . . 25
vi
4 Cubic Planar Graphs
27
4.1
2-Edge-Connected Cubic Planar Graphs . . . . . . . . . . . . . . . . . . . . . 27
4.2
General Cubic Planar Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
4.2.1
Structure of the Graph . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
4.2.2
Edge Weightings of the Graph . . . . . . . . . . . . . . . . . . . . . . 29
4.2.3
Reference Star Labelling of the Graph . . . . . . . . . . . . . . . . . . 32
4.2.4
Arbitrary Star Labellings of the Graph . . . . . . . . . . . . . . . . . . 34
4.2.5
Signs of the Star Labellings of the Graph . . . . . . . . . . . . . . . . 39
4.2.6
Proof of Theorem 1.0.4 . . . . . . . . . . . . . . . . . . . . . . . . . . 42
Bibliography
46
vii
List of Figures
1.1
A graph G and its line graph L(G). . . . . . . . . . . . . . . . . . . . . . . . .
2
2.1
A graph with two star labellings consistent with the edge weighting W . . . .
5
3.1
The edge weighting W G of a graph G with a single cycle in its 2-factor. . . .
9
3.2
One of the two star labellings that are consistent with
W G,F
of a cubic graph
G with a single cycle in its 2-factor. . . . . . . . . . . . . . . . . . . . . . . .
9
3.3
The edge weights assigned to an unpaired cycle and its connecting edges. . . 12
3.4
The preliminary labels for an unpaired cycle. . . . . . . . . . . . . . . . . . . 12
3.5
The unique star labelling for an unpaired cycle. . . . . . . . . . . . . . . . . . 13
3.6
The edge weights assigned to a paired cycle and its connecting edges. . . . . . 14
3.7
The unique star labelling for a paired cycle. . . . . . . . . . . . . . . . . . . . 15
3.8
An example of a graph G with its tree T and graphs H ∈ Comp(G \ B) with
the corresponding H ∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.9
The edge weights for H ∈ Comp(G \ B) of Type(1) and Type(2). . . . . . . . 19
3.10 The edge weights for threads of H ∈ Comp(G \ B) of Type (3). . . . . . . . . 20
3.11 The subgraph Cb of G in Lemma 3.2.2. . . . . . . . . . . . . . . . . . . . . . . 22
3.12 The unique star labelling for H ∈ Comp(G \ B) of Type (1) or Type (2). . . . 23
3.13 The unique star labellings for threads. . . . . . . . . . . . . . . . . . . . . . . 24
4.1
The primary and secondary edge weights for the edges of threads of
H ∈ Comp(G \ B) of Type (3). . . . . . . . . . . . . . . . . . . . . . . . . . . 30
4.2
The edge weighting W∅ of the example graph from Figure 3.8. . . . . . . . . . 31
4.3
The star labelling ρ for H ∈ Comp(G \ B). . . . . . . . . . . . . . . . . . . . 33
4.4
The reference star labelling ρ for the example graph from Figure 3.8. . . . . . 35
viii
4.5
The general star labelling π for a blue thread of H ∈ Comp(G \ B)
of Type(3). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4.6
The general star labelling π for red or green threads of H ∈ Comp(G \ B) of
Type (3). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
4.7
Applying secondary even (option 2) case labels to a thread of length m ≥ 2
leads to a contradiction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
ix
Chapter 1
Introduction
We consider a graph G = (V, E), possibly having multiple edges, but with no loops. Basic
graph theory definitions can be found in West’s Introduction to Graph Theory [15]. Given a
set of colours C, a list assignment for the edges of G is L = {Le }e∈E , where Le ⊂ C. For
a list assignment L, a L-edge-colouring of G is a proper edge-colouring c : E → C of G
where c(e) ∈ Le . Similarly, we can define a list assignment for vertices, L = {Lv }v∈V , and
a proper colouring of the vertices that uses colours from these lists is an L-colouring.
We say G is k-edge-choosable for a vector k = (ke )e∈E ∈ NE if G is L-edge-choosable,
for every list assignment L such that |Le | ≥ ke for each edge e ∈ E. The smallest integer
k such that G has a k-edge-colouring is the chromatic index, χ0 (G) = k. The smallest
integer k such that G is (k, . . . , k)-edge-choosable is the list chromatic index, χ0` (G) = k.
It is obvious that χ0 (G) ≤ χ0` (G), since the edge list-colouring where each edge is assigned the list {1, 2, . . . χ0` (G)} gives a proper χ0 (G)-edge-colouring of G. In fact, the Listcolouring Conjecture states that χ0 (G) = χ0` (G) for any graph G. Jensen and Toft’s
Graph Colouring Problems [11] cites Häggkvist and Chetwynd [8], who assert that several
people thought of this conjecture independently, including V.G. Vizing, R.P. Gupta, and
M.O. Albertson and K.L. Collins but that it was first published by Bollobás and Harris in
1985 [4]. This is a difficult conjecture to prove; it has been verified for several classes of
graphs including, among others, bipartite graphs [7], 1-factorable planar graphs [6], and Kn
where n is odd [9]. The last two of these results were proven using the polynomial method,
which is described in Section 2.
In our case, the polynomial of interest for the polynomial method is the graph monomial
(we are following the terminology of [6]).
1
CHAPTER 1. INTRODUCTION
2
G
L(G)
Figure 1.1: A graph G and its line graph L(G).
Definition Let x = (xv )v∈V . The graph monomial of G = (V, E), pG (x), is
pG (x) =
Y
(xu − xv )µ(u,v)
uv∈E
where µ(u, v) is the number of edges of E joining u, v ∈ V .
Let L = {Lv }v∈V be a list assignment with Lv ⊂ Z for each vertex. Then, we see that
G has an L-colouring if and only if there exist sv ∈ Lv for each vertex v ∈ V , such that
pG ((sv )v∈V ) 6= 0.
We define the line graph, L(G), of G = (V, E). The set of vertices of L(G) is the set
of edges of G, E. Two vertices of L(G), e1 , e2 ∈ E, are connected by the same number of
edges in L(G) as they have common endpoints in G. For example, two parallel edges of
G become two vertices of L(G) joined by parallel edges. Thus, list edge-colourings of G
correspond to list colourings of L(G). Combining the graph monomial with the line graph
gives us an algebraic test for the existence of a list edge-colouring of G: for a list assignment
L = {Le }e∈V , G has a L-edge-colouring if and only if there exist se ∈ Le for each edge e ∈ E
such that pL(G) ((se )e∈E ) 6= 0.
A Brooks’ type theorem combines with the line graph to give us our first result about
edge-choosability.
Theorem 1.0.1 (Brooks’ Theorem for Choosability [14]) Every loopless connected
graph G that is neither a complete graph nor an odd cycle is ∆(G)-choosable.
CHAPTER 1. INTRODUCTION
3
Since the line graph of a cubic graph is 4-regular but is not the complete graph on 5 vertices,
the following corollary is immediate.
Corollary 1.0.2 If G is a cubic graph, then χ0` (G) ≤ 4
Later, we will see a result of Ellingham and Goddyn [6] that implies that if χ0 (G) = 3
for a planar cubic graph G, then χ0 (G) = 3 = χ0` (G). So Corollary 1.0.2 implies that the
list-colouring conjecture is true for planar cubic graphs.
In this thesis, we consider the edge-choosability cubic graphs G using the polynomial
method. From Corollary 1.0.2, we already know that χ0` (G) ≤ 4. We will reprove this result
and give tighter bounds on the list sizes, which are particularly strong in the case where G
is also planar.
In Chapter 2, we introduce the Combinatorial Nullstellensatz, the polynomial method
and star labellings of a graph, and we show how they relate to list edge-colourings of a
graph. Chapter 3 and Chapter 4 each prove one the following theorems.
Theorem 1.0.3 If G is a cubic graph, then χ0` (G) ≤ 4.
Note that Theorem 1.0.3 is the same as Corollary 1.0.2. Not only do we use a different
proof method, but our proof shows that about
2
3 |E(G)|
edges use smaller list size, and
it also gives a stronger result about the coefficients of the line graph monomial of G (see
Section 2). When G is planar, we get a stronger result.
Theorem 1.0.4 If G is a planar cubic graph with c cut edges, then G is k-edge-choosable
for some k = (ke )e∈E(G) ≤ (4, 4 . . . , 4), where ke = 4 holds for at most 3c edges.
Chapter 2
The Polynomial Method and
Star Labellings
The polynomial method is based on the Combinatorial Nullstellensatz; a proof of the following theorem can be found in [2].
Theorem 2.0.5 (Combinatorial Nullstellensatz) Let F be a field, and let
f = f (x1 , x2 , . . . , xn ) be a polynomial in F[x1 , x2 , . . . , xn ]. Suppose that deg(f ) =
where each ti is a nonnegative integer and that the coefficient of
n
Y
n
X
ti ,
i=1
xtii in f is nonzero. If
i=1
S1 , S2 , . . . , Sn are subsets of F with |Si | > ti , then there exist s1 ∈ S1 , . . . sn ∈ Sn so that
f (s1 , . . . , sn ) 6= 0.
In general, the polynomial method is the application of the Combinatorial Nullstellensatz to a given polynomial to prove “list related” properties of objects such as graphs,
hypergraphs and groups. Alon and Tarsi pioneered this method in [1] and applications of
this method to many different areas are collected in [3].
Given a graph G, we will apply the polynomial method to find a bound on the list
edge-choosability of G by interpreting the coefficients of pL(G) (x) combinatorially using star
labellings of the graph. The following definitions are based on [6].
A star labelling πv of a vertex v ∈ V is a bijection πv : δ(v) → {0, 1, . . . , deg(v) − 1},
where δ(v) is the set of edges incident with v. If πv (e) = k, then we say v is incident to the
label k at the edge e. A star labelling π of G is a function that assigns a star labelling
4
CHAPTER 2. THE POLYNOMIAL METHOD AND STAR LABELLINGS
2
1
2
0
2
1
0
5
1
1
2
3
0 2
0
5
0
2
1
0
2
0
2
1
1
1
3
5
2
0 2
0
5
3
0
3
1
1
2
2
0
1
3
Figure 2.1: A graph with two star labellings (centre and right) consistent with the edge
weighting W (left).
to each vertex of G, that is π : v 7→ πv . For a given graph, we fix a star labelling ρ to
be the reference labelling of the graph. We let S[n] be the permutation group of the set
{0, . . . , n − 1}. For every vertex, we define the sign of a star labelling of a vertex to be
−1
sgn(πv ) = sgn(πv ◦ ρ−1
v ), since πv ◦ ρv ∈ S[deg(v)] . Then, the sign of a star labelling of
Y
the graph G is sgn(π) =
sgn(πv ).
v∈G
Star labellings of a graph G are closely associated with edge weightings, W : E → ZE .
A star labelling is consistent with an edge weighting W if for all e = uv ∈ E,
πu (e) + πv (e) = W (e). Figure 2.1 shows a graph G with an edge weighting W (left)
and two star labellings of G consistent with W (centre and right). If we take the centre
star labelling to be the reference labelling ρ, then the right labelling has negative sign, since
the labelling of each of the three vertices incident with the outer face differs from ρ by one
transposition, and the star labelling of the fourth vertex is the same.
The graph monomial of a graph G is of uniform degree, so if any term α
Y
xtee
e∈E(G)
of pL(G) (x) has nonzero coefficient, then G is (te + 1)e∈E -edge-choosable. We prove the
following theorem by relating the coefficients of the graph monomial of L(G) to the star
labellings of G.
CHAPTER 2. THE POLYNOMIAL METHOD AND STAR LABELLINGS
6
Theorem 2.0.6 Consider a graph G = (V, E) with an edge weighting W . If the number
of star labellings of G of positive sign consistent with W is not equal to the number of
star labellings of negative sign consistent with W , then G is (W + 1)-edge-choosable, where
(W + 1) = (W (e) + 1)e∈E .
Proof Suppose an edge weighting W satisfies the hypothesis. We will show that the coefY
ficient of
xeW (e) in pL (G)(x) is
e∈E
±
X
{sgn(π) : π is a star labelling of G consistent with W }.
Then our hypothesis tells us that this coefficient is non-zero, and so the result holds by
the Combinatorial Nullstellensatz.
We consider the graph monomial of the line graph of G, and we fix a reference star
0
labelling ρ of G. For u ∈ V and 0 ≤ i ≤ deg(v) − 1, we define e(u,i) = ρ−1
u (i) = e ∈ E such
that ρu (e0 ) = i. We denote the number of common end points of e0 , e00 ∈ E by ν(e0 , e00 ).
Y
0 00
{(xe0 − xe00 )ν(e ,e ) : e0 6= e00 ; e0 , e00 ∈ E}
Y Y
=±
{(xe0 − xe00 ) : e0 6= e00 , e0 and e00 ∈ E are incident with u}
pL(G) (x) =
u∈V
=±
Y Y
(xe(u,i) − xe(u,j) ) .
u∈V
i<j
For each u ∈ V the product is a Vandermonde determinant,
Y
pL(G) (x) = ±
det({xie(u,j) }i,j∈0,...deg(u)−1 )
u∈V
=±
Y
X
u∈V (G) σ∈S[deg(u)]
σ(1)
σ(deg(u)−1)
sgn(σ)xσ(0)
e(u,0) xe(u,1) · · · xe(u,deg(u)−1) .
CHAPTER 2. THE POLYNOMIAL METHOD AND STAR LABELLINGS
7
The set of permutations S[deg(u)] is {πu ◦ ρ−1
u : πu is a star labelling of u}.
If πu ◦ ρ−1
u = σ ∈ S[deg(u)] , then sgn(πu ) = sgn(σ). Thus,
Y
X
πu (e
) πu (e
)
πu (e(u,deg(u)−1) )
pL(G) (x) = ±
sgn(πu )xe(u,0)(u,0) xe(u,1)(u,1) · · · xe(u,deg(u)−1)
u∈V (G) star labellings πu
X
=±
sgn(π)
star labellings π of G
=±
X
X
Y
xeπu (e)+πv (e)
e=uv∈E
Y
0 (e)
{sgn(π) : π is a star labelling consistent with W }
xW
e
W 0 ∈ZE
0
!
.
e∈E
So to show edge-choosability for a graph G, we can find an edge weighting W that is
consistent with unequal numbers of positive and negative star labellings. One easy way to
do this is to choose an edge weighting where all consistent star labellings have the same
sign. This is the approach used in the proof of Theorem 1.0.3. A second approach, used in
the proof of Theorem 1.0.4, is to consider multiple edge weightings of G.
Corollary 2.0.7 Consider a graph G = (V, E) and a set W of edge weightings of G. If
the number of star labellings of G of positive sign consistent with any W ∈ W is not equal
to the number of star labellings of negative sign consistent with any W ∈ W, then G is
(W 0 + 1)-edge-choosable for at least one W 0 ∈ W. Also, G is (Wmax + 1)-edge-choosable,
where Wmax (e) = max{W (e) : W ∈ W}.
Proof For at least one W 0 ∈ W the number of star labellings of G of positive sign consistent
with W 0 is not equal to the number of star labellings of negative sign consistent W 0 . By
Theorem 2.0.6, G is (W 0 + 1)-edge-choosable and W 0 ≤ Wmax , so G is
(Wmax + 1)-edge-choosable.
Chapter 3
Cubic Graphs
In this chapter, we consider any connected cubic graph G and prove Theorem 1.0.3.
3.1
2-Edge-Connected Cubic Graphs
We first consider the case where G is 2-edge-connected. Then G has 2-factor [15, p.139].
Let F be a 2-factor of G with k component cycles. We denote the set of cycles of F by F.
In Section 3.1.1 and Section 3.1.2, we will use this 2-factor F to define an edge weighting
W G,F of G and explore the star labellings that are consistent with W G,F .
3.1.1
Hamiltonian 2-Factor
First, we consider the case where the 2-factor F consists of a Hamiltonian cycle, so k = 1.
Then the graph G is an even cycle C = v1 e1 v2 . . . vm em v1 of length m with
define the edge weighting
m
2
chords. We
W G,F
by,

3 for e = ei ∈ E(C), i = 1 . . . m
W G,F (e) =
0 for e = c, a chord of G
(3.1)
as illustrated in Figure 3.1.
Lemma 3.1.1 A 2-edge-connected cubic graph G with a 2-factor consisting of a single cycle
has exactly two star labellings consistent with W G,F as defined in (3.1), and these star
labellings have the same sign.
8
CHAPTER 3. CUBIC GRAPHS
9
3
3
0
3
0
3
3
0
3
v2
3
0 e1
v1
3
em
vm
Figure 3.1: The edge weighting W G of a graph G with a single cycle in its 2-factor.
3 1 2
0
2
3
1 0
0
3
1 2 3
0
1 v2
0 2
3
0 0 1
0
0 v1
2
3
1
0
2 vm
0
3
1
3 2
Figure 3.2: One of the two star labellings that are consistent with W G,F of a cubic graph
G with a single cycle in its 2-factor.
CHAPTER 3. CUBIC GRAPHS
10
Proof Any star labelling π consistent with W G,F must have the labels πvi (c) = 0 at each
vi ∈ V (C) where c is the chord with endpoint vi . Since π is a star labelling, one of the
labels πvi (ei−1 ) and πvi (ei ) is 1 and the other label is 2. One possible star labelling is seen
in Figure 3.2. Since the star labelling π is consistent with W G,F , π is one of two possible
star labellings, π1 and π2 , which are defined for each vertex vi for i = 1, . . . m by,


0 for e = c, the chord with endpoint vi



(π1 )vi (e) = 1 for e = ei−1




2 for e = ei


0 for e = c, the chord with endpoint vi



(π2 )vi (e) = 2 for e = ei−1




1 for e = ei
Since these labellings differ by a transposition at each vertex, sgn((π1 )vi ) = − sgn((π2 )vi )
for i = 1, . . . , m and so
sgn(π1 ) =
Y
v∈V (G)
((π1 )v ) =
Y
((π2 )v ) = sgn(π2 )
v∈V (G)
since |V (G)| is even.
3.1.2
2-Factor Containing Multiple Cycles
Suppose instead that the 2-factor F has k ≥ 2 cycles. We say two cycles C1 , C2 ∈ F, are
adjacent if there is a connecting edge of C1 , uv ∈ E(G), with u ∈ V (C1 ) and v ∈ V (C2 ).
We also say uv is a connecting edge of u and v. We fix a maximal pairing of adjacent cycles.
We let P ⊂ F be the set of paired cycles of F and U = F \ P be the set of unpaired cycles.
We note that because the pairing is maximal, each unpaired cycle C ∈ U is adjacent only
to cycles in P.
For each cycle C ∈ P, we designate a connecting edge between C and its pair C 0 as the
link edge of both C and C 0 , cC = cC 0 . For a cycle C ∈ U, we pick any connecting edge of
C to be its link edge, cC . For each C = v1 e1 . . . vm em ∈ F, we may assume that v1 ∈ V (C)
is the endpoint of cC . Then v1 is the base vertex of C and we designate the edge e1 as the
base edge of C.
CHAPTER 3. CUBIC GRAPHS
We can now define the edge



2






2



W G,F (e) = 1





3





0
11
weighting W G,F by,
for e = e1 is a base edge in some C ∈ F
for e = cC is a link edge for some C ∈ P
for e = cC is a link edge for some C ∈ U
(3.2)
for e = ei is a non-base edge in some C ∈ F
for all other edges e of G
The following star labelling of G, ρ, is consistent with W G,F . The star labelling ρ is
illustrated in Figure 3.5 and Figure 3.7.
For each cycle



0





2
ρvi (ej ) =


1





2
C = v1 e1 . . . em v1 ∈ F
for i = 1, j = 1
for i = 2, j = 1
(3.3)
for i = j 6= 1
for i 6= 2, i = j + 1
mod m
ρv1 (cC ) = 1
(3.4)
ρvi (c) = 0 for all cord edges and connecting edges c 6= cC with endpoint vi
(3.5)
The two following lemmas, Lemma 3.1.2 and Lemma 3.1.3, show that ρ is the unique star
labelling of G that is consistent with W G,F . In the former, we consider the star labellings
of the vertices of the unpaired cycles, and in the latter, we consider the star labellings of
the vertices of the paired cycles.
Lemma 3.1.2 Let G be a 2-edge-connected cubic graph with a 2-factor F of G of at least
2 cycles. If the star labelling π of G is consistent with W G,F as defined in (3.2), then we
have πv = ρv for every vertex v of each C ∈ U. In particular, the label of the base vertex at
the link edge is πv1 (cC ) = 1.
Proof We recall that W G,F assigns edge weights as shown in Figure 3.3. Each edge c that
is a chord or connecting edge of F with endpoint vi for i 6= 1 (so c is not cC ), has edge
weight W (c) = 0. So the label must be πvi (c) = 0 = ρvi (c). For the link edge of C, eC , we
recall that W G,F (cC ) = 1, and we let δ = πv1 (cC ) so δ ∈ {0, 1}, as shown in Figure 3.4.
CHAPTER 3. CUBIC GRAPHS
12
0
0
3
3
3
0
3
v2
e1
2
0
cC
v1 1
3 em
3
3
vm
0
Figure 3.3: The edge weights assigned to an unpaired cycle and its connecting edges. The
base and link edges are in bold.
0
0
3
0
3
0
3
0
3
0
0
3
3
0
0
v2
e1
2
δ cC
v1 1
3 e
m
vm
0
0
Figure 3.4: The preliminary labels for an unpaired cycle.
CHAPTER 3. CUBIC GRAPHS
13
0
0 2
1
1
2 3
3
0
0
10 2
0
v2
2
e1
2 0
1 cC
0
v1
2 1
3 em
0
1
v
3
2 m
3
1
2 0
3
3
10
0
Figure 3.5: The unique star labelling for an unpaired cycle.
For all non-base cycle edges, ei ∈ E(C) i 6= 1, the edge weight is W G,F (ei ) = 3, and the
edge weight of the base edge e1 is W G,F (e1 ) = 2. Since all vertices are incident to the labels
0, 1 and 2, summing the labels over all vertices of the cycle we have,


X

W G,F (e) + δ + 0 + · · · + 0 = (0 + 1 + 2)|V (C)|
e∈E(C)
3 + 3 + · · · + 3 + 2 + δ = 3|V (C)|,
and so πv1 (cC ) = δ = 1 = ρv1 (cC ). Since en has edge weight W G,F (en ) = 3, the other
labels of v1 are πv1 (en ) = 2 = ρv1 (en ) and πv1 (e1 ) = 0 = ρv1 (e1 ). These labels are shown as
the boxed labels in Figure 3.5. Then, as the base edge e1 has edge weight W G,F (e1 ) = 2,
we know πv2 (e1 ) = 2 = ρv2 (e1 ). We know v2 is incident with the label 0 at its chord or
connecting edge, so the final label incident with v2 is πv2 (e2 ) = 1 = ρv2 (e2 ). Similarly, we
see that the remaining labels for the vertices in V (C) are πvi (ei−1 ) = 2 = ρvi (ei−1 ) and
πvi (ei ) = 1 = ρvi (ei ) for i = 3, . . . , n, as shown in Figure 3.5. So πv = ρv for all v ∈ V (C).
CHAPTER 3. CUBIC GRAPHS
0
3
14
1
3
3
3
0
v2
2
v1
0
3
3
3
0
0
0
0
3
e1
em
vm
3
2
cC
2
1
3
3
3
0
3
3
1
Figure 3.6: The edge weights assigned to a paired cycle and its connecting edges. The link
and base edges are in bold.
Lemma 3.1.3 Let G be a 2-edge-connected cubic graph with a 2-factor F of G of at least 2
cycles. If the star labelling π of G is consistent with W G,F as in (3.2), then we have πv = ρv
for every vertex v of each C ∈ P. In particular, the label of the base vertex at the link edge
is πv1 (cC ) = 1.
Proof We recall that the edge weighting W G,F is as shown in Figure 3.6. We consider
any connecting edge c of C with endpoint vi ∈ V (C) that is not the link edge cC . The
connecting edge c may be a link edge of some unpaired cycle and so have edge weight
W G,F (c) = 1, or else c has edge weight W G,F G(c) = 0. In both cases the labelling at vi
must have πvi (c) = 0 = ρvi (c). In the second case, this label follows since the endpoint u of
c in the unpaired cycle must have the label πu (c) = 1 by Lemma 3.1.3.
By the same double counting argument as in the proof of Lemma 3.1.2, we see that any
star labelling of V (C) consistent with W G,F must have the label πv1 (cC ) = 1 = ρv1 (cC ).
Again, as in the proof of Lemma 3.1.2, since en has edge weight W G,F (en ) = 3, the other
labels of v1 are πv1 (en ) = 2 = ρv1 (en ) and πv1 (e1 ) = 0 = ρv1 (e1 ), these labels are shown as
CHAPTER 3. CUBIC GRAPHS
0
0 2
1
3
2
3
1 0
3
1
0
1 2
0
0
3 1 2
v2
e1
2 0
1
0
v1
2
3 e
1 m
0
v
3
2 m
3
1
0
0
15
0
cC
2
1
2
01
0
0
2 1
3
0
3
2 01
3
2
2
0
2
1
0
0
1
0
1
3
3
2
3
3
1
0 2
1
Figure 3.7: The unique star labelling for a paired cycle.
the squared labels in Figure 3.7. Since the base edge e1 has edge weight W G,F (e1 ) = 2, we
know πv2 (e1 ) = 2 = ρv2 (e1 ). We know v2 is incident with the label 0 at its connecting edge so
the final label incident with v2 is πv2 (e2 ) = 1 = ρv2 (e2 ). Similarly, we see that the remaining
labels for the vertices in V (C) are πvi (ei−1 ) = 2 = ρvi (ei−1 ) and πvi (ei ) = 1 = ρvi (ei ) for
i = 3, . . . , n, as seen in Figure 3.7. So πv = ρv for all v ∈ V (C).
Corollary 3.1.4 Let G be a 2-edge-connected cubic graph and F a 2-factor of G of at least
2 cycles. Then there is a unique star labeling π of G consistent with W G,F as defined in
(3.2).
Proof This follows immediately from Lemma 3.1.2 and Lemma 3.1.3.
Lemma 3.1.5 Any 2-edge-connected cubic graph G is 4-edge-choosable.
Proof Let F be a 2-factor of G with k cycles, and let W G,F be the corresponding edge
weighting as defined in (3.1) if k = 1 or in (3.2) if k ≥ 2. If F consists of a single cycle,
then by Lemma 3.1.1, there are exactly two star labellings of G consistent with W G,F , and
both have the same sign. Otherwise, Lemma 3.1.4, there is exactly one star labelling of G
CHAPTER 3. CUBIC GRAPHS
16
consistent with W G,F . In both cases, there are a different number of star labelling with
positive and negative sign. By Theorem 2.0.6, G is 4-edge-choosable since W G,F ≤ 3.
3.2
General Cubic Graphs
Now we consider any connected cubic graph G.
3.2.1
Structure of the Graph
We let B be the set of cut edges of G, so G \ B consists of 2-edge-connected components and
isolated vertices, and we denote the set of these components by Comp(G \ B). The vertices
of a subgraph H ∈ Comp(G \ B) have degrees 0, 2 or 3, and H must have an even number
of odd degree vertices. Thus, H must be of one of the following types:
Type (1) a single vertex,
Type (2) a cycle, or
Type (3) a graph with at least two vertices of degree 3.
Let T be the tree formed from G by contracting each subgraph H ∈ Comp(G \ B) to a
single vertex. Then, T has edges E(T ) = B and vertices V (T ) = {H : H ∈ Comp(G \ B)}.
We root T at a leaf Hr . The root Hr must be of Type (3), since for any vertex H of T
of Type (1) or Type (2), H must be adjacent to more than one cut edge in T . For each
H 6= Hr , consider the cut edge bH ∈ B that is above it in T (which is incident to H on the
path from Hr to H in T ). The endpoint of bH that lies in V (H) is the root vertex of H.
Other edges of B that are incident with H in T (and G) are referred to as the bridge edges
below H, and each such bridge edge is below the vertex v ∈ V (H) that is its endpoint.
We denote the set of bridge edges below H by B(H), and the bridge edge below a vertex v
by B(v).
Consider an H of Type (3).
Any maximal path t = v1 e1 . . . vm em vm+1 of length
m ≥ 2 and such that all internal vertices have degree 2 is called a thread of H. We
let Vin (t) = {v2 , . . . vm } be the set of internal vertices of t. Construct the graph H ∗ by
suppressing all internal vertices of the threads of H, that is, by replacing every thread
t with a single edge et = v1 vm+1 . Each edge et in H ∗ that corresponds to a thread t in
H is called a thread edge. All edges of E(H) ∩ E(H ∗ ) are referred to as non-thread
CHAPTER 3. CUBIC GRAPHS
17
edges. Since the root vertex of H 6= Hr has degree 2, it is in a thread. We designate the
corresponding thread edge as the root edge of H ∗ . Let
G∗ = ∪ {H ∗ : H ∈ Comp(G \ B), H is of Type (3)}.
For each H ∈ Comp(G \ B), we will distinguish certain vertices and edges of H and
B(H). If H is of Type (1), then H is a single vertex, which we name v1 , and we name
the two cut edges in B(H) b1 and b2 . If H is of Type (2), then we let the cycle H be
v1 e1 v2 e2 . . . vn en v1 , where v1 is the root vertex. We let the edges in B(H) be bi = B(vi ),
for i = 2, . . . , n. If H is of Type (3), then each thread t is t = v1 e1 . . . vm em vm+1 , and we
let bi = B(vi ) for i = 2, . . . m. If some vj ∈ V (t) is the root vertex of H, then bj does not
S
exist. Let B(t) = v∈V (t) B(v) be the set of cut edges below the thread t.
3.2.2
Edge Weighting of the Graph
We define an edge weighting W of G by the following procedure, which assigns edge weights
to the edges E(H) ∪ B(H) for each H ∈ Comp(G \ B). If H is of Type (1), then we assign
the edge weights W (b1 ) = 3 and W (b2 ) = 1. If H is of Type (2), then we assign the edge
weights W (e1 ) = 3 and W (b2 ) = 1. For every other edge f ∈ (E(H) ∪ B(H)) \ {e1 , b2 },
the edge weight is W (f ) = 2. These edge weights are illustrated in Figure 3.9. For both
Type (1) and Type (2), the average edge weight over E(H) ∪ B(H) is 2.
Finally, we consider an H ∈ Comp(G \ B) of Type (3). This includes the case where
H = Hr . Each such H corresponds to some H ∗ . In the non root case, each H ∗ is
2-edge-connected and cubic and has a root edge e. We can find a perfect-matching of H ∗
∗
that includes e [12]. The complement of this perfect matching is a 2-factor F H . The
root edge, e, is either a chord of a cycles C ∈ F or a connecting edge between two cycles
C1 , C2 ∈ F. If e is a connecting edge, then we take a maximal pairing that contains the pair
C1 , C2 . If e is a chord we take any maximal pairing. For the root Hr , we take any 2-Factor
∗
F Hr of Hr∗ and any maximal pairing of cycles. Either way, we define the edge weighting
WH
∗ ,F H ∗
as in (3.1) or (3.2) and fix a star labelling π ∗ of H ∗ consistent with W H
We assign each non-thread edge
e0
∈ E(H) ∩
E(H ∗ )
the weight
W (e0 )
=
∗ ,F H ∗
∗
H
W H ,F
∗
.
(e0 ).
We consider each thread t = v1 e1 v2 . . . en vm+1 of H and the corresponding thread edge
et = v1 vm+1 of H ∗ , and assign edge weights to the edges E(t) ∪ B(t) according to the
procedure below which depends on the labels πv∗1 (et ) and πv∗m+1 (et ). We may assume that
πv∗1 (e1 ) ≤ πv∗m+1 (em ). There are five cases, denoted Case(s,t) where s = πv∗1 (et ) and
t = πv∗m+1 (et ), so s ≤ t. The resulting edge weights are illustrated in Figure 3.10.
CHAPTER 3. CUBIC GRAPHS
18
i
G
1
H2
H1
Type
Hi
Hi*
(3)
3,6, (3)
9,10
H6
H3
H4
H5
H7
H8
2
(1)
4
(1)
5
(2)
7
(3)
8
(3)
H9
NA
NA
H10
T
H1
H2
H6
H4
H5
H3
H8
H7
H10
Root vertices and edges are in bold.
H9
Figure 3.8: An example of a graph G with its tree T and graphs H ∈ Comp(G \ B) with
the corresponding H ∗ .
CHAPTER 3. CUBIC GRAPHS
19
Type (2):
v1 e1 v2 b2
2
2
vn
b1 b2
3
2
2 2
1
3
v1
1
Type (1):
2
Figure 3.9: The edge weights for H ∈ Comp(G \ B) of Type(1) and Type(2).
Case(1,2): We assign the edge weights W (e1 ) = 1, W (e2 ) = 3 and W (b2 ) = 3. We
assign the edge weight W (f ) = 2 to the remaining edges f ∈ (E(t) ∪ B(t)) \ {e1 , e2 , b2 }.
Case(1,1): We assign the edge weights W (e1 ) = 1 and W (em+1 ) = 3. We assign edge
weight W (f ) = 2 to the remaining edges f ∈ (E(t) ∪ B(t)) \ {e1 , em+1 }.
Case(0,2): We assign the edge weights W (e1 ) = 0, W (e2 ) = 3 and W (b2 ) = 3. We
assign the edge weight W (f ) = 2 to the remaining edges f ∈ (E(t) ∪ B(t)) \ {e1 , e2 , b2 }.
Case(0,1): We assign the edge weights W (em ) = 3, W (em−1 ) = 1 and W (bm ) = 1.
We assign edge weight W (f ) = 2 to the remaining edges f ∈ (E(t)∪B(t))\{em , em−1 , bm }.
Case(0,0): We assign the edge weights W (e1 ) = 0. We assign edge weight W (f ) = 2
to the remaining edges f ∈ (E(t) ∪ B(t)) \ {e1 }.
We note that a root thread t corresponds to a root edge et , which is either a link
edge connecting paired cycles with weight W H
∗ ,F H ∗
(et ) = 2 and labels πv∗1 (et ) = 1 and
πv∗m+1 (et ) = 1 (by Lemma 3.1.3), or chord of weight W H
∗ ,F H ∗
(et ) = 0 with labels πv∗1 (et ) = 0
and πv∗m+1 (et ) = 0. That is et is either in Case(1,1) or Case(0,0) when t is a root thread.
Lemma 3.2.1 For any H ∈ Comp(G \ B), the average edge weight W (e) over the edges in
E(H) ∪ B(H) is 2.
Proof We have already seen this for H of Type (1) or Type (2) at the beginning of this
∗
section. For H of Type (3) with 2-factor F H as chosen above, we can see from Figure 3.10
CHAPTER 3. CUBIC GRAPHS
Case(1,2):
3
1
et
v1
Case(1,1):
1
v1
2
et
v1 1
2
0
v1
et
v1
vm+1
v1
0
vm+1
v2 3
b2 3
0
e1
b2 3
2
vm+1
em
2
b2 2
v2
2
e2
v2
e1
vm+1
Case(0,1):
1
1
et
v
0
v1 1
1
2
Case(0,0):
0
e1
vm+1
Case(0,2):
2
0
e
t
v1
m+1
20
bm 2
2
3
e2
vm+1 3 vm 1
em
em-1
bm 1
v1 0 v2 2
e1
e2
b2 2
2
2
2
2
2
2
3 vm+1
em
vm+1
em
v1
e1
vm+1
em
Figure 3.10: The edge weights for threads of H ∈ Comp(G \ B) of Type (3). Note that the
middle section of each diagram is repeated 0 or more times to make a thread of length m.
.
CHAPTER 3. CUBIC GRAPHS
21
that the total edge weight for each thread t of H, with corresponding thread edge et , is
WH
∗ ,F H ∗
(et ) + 2|E(t) + B(t)| − 2. We note that the cubic graph H ∗ has a star labelling
that is consistent with the edge weighting W H
in Section 3.1.1, if F
H∗
∗ ,F H ∗
. One such star labelling is given by π1
∗
has only one cycle, or by ρ in Section 3.1.2, if F H has multiple
cycles. Thus we know that
X
WH
∗ ,F H ∗
(e) = (0 + 1 + 2)|V (H ∗ )|
e∈E(H ∗
and by the handshake theorem,
X
2
∗
H∗
W H ,F (e) = 3( |E(H ∗ )|)
3
∗
e∈E(H
= 2|E(H ∗ )|.
So the sum of the edge weights in E(H) ∪ B(H) is,
X
X
W (e) =
e∈E(H)∪B(H)
et
+
=
WH
∗ ,F H ∗
(e)
6 e∈E(H ∗ )
=
X
∗
H∗
{W H ,F (et ) + 2|E(t) + B(t)| − 2 : t is a thread of H}
X
X
∗
H∗
W H ,F (e) +
{2|E(t) + B(t)| − 2 : t is a thread of H}
e∈E(H ∗ )
X
= 2 |E(H ∗ )| +
{|E(t) + B(t)| − 1 : t is a thread of H} .
The set of edges E(H)∪B(H) is the set of edges E(H ∗ ) with each thread edge et replaced
by the edges E(t) ∪ B(t). Then,
|E(H) ∪ B(H)| = |E(H ∗ )| +
X
{|E(t) + B(t)| − 1 : t is a thread of H}.
Thus the average edge weight over E(H) ∪ B(H) is 2.
3.2.3
Star Labellings of the Graph
We define the lower endpoint of a cut edge b ∈ B, low(b), to be the endpoint of b farthest
from Hr in T ; the other (nearer) endpoint of b is the upper endpoint, up(b). We say a
star labelling π is balanced if for every cut edge b, πlow(b) (b) = 1.
CHAPTER 3. CUBIC GRAPHS
22
b
α
C
Figure 3.11: The subgraph Cb of G in Lemma 3.2.2.
Lemma 3.2.2 Any star labelling π of G that is consistent with the edge weighting W defined
in Section 3.2.2 is balanced.
Proof Consider any cut edge b ∈ B and the component Cb of G \ b that contains low(b).
By Lemma 3.2.1, the average edge weight over E(H) ∪ B(H) for each H ∈ G \ B is 2. Thus,
the average edge weight over E(Cb ) is 2. Let πlow(B) (b) = α be the lower label of b as in
Figure 3.11. The sum of labels around a vertex is 0 + 1 + 2 = 3, and the sum of labels for
any edge is its weight. So summing the labels of π in G around every vertex in V (Cb ) gives,
X
3|V (Cb )| = α +
WS (e) = α + 2|E(Cb )|.
e∈E(Cb )
Now, |V (Cb )| is odd since Cb must have an even number of degree 3 vertices and one vertex
of degree 2. So α ∈ {0, 1, 2} must be odd, that is α = 1. Thus π is balanced.
As an immediate consequence of Lemma 3.2.2, we see that if b ∈ B is a cut edge of G,
then any star labelling π of G that is consistent with W must have the labels πlow(b) (b) = 1
and πup(b) (b) = W (b) − 1.
Lemma 3.2.3 All star labellings of G that are consistent with W defined in Section 3.2.2
have the same sign.
Proof We consider each H ∈ Comp(G \ B) in turn and show that the overall sign in V (H),
Y
πv , is the same for all possible star labellings π of G consistent with W .
v∈V (H)
CHAPTER 3. CUBIC GRAPHS
23
Type (1):
Type (2):
2
2
0
1
2 1 1
2
(a)
2
2
0
e1 1
1
0 b2
2
1
2
1
0
1
3
2
2 0
b1 b2
1
3
1
v1 1
2
v1 1
(b)
Figure 3.12: The unique star labelling for H ∈ Comp(G \ B) of Type (1) or Type (2).
If H is of Type (1), then the labelling at v1 must be πv1 (bH ) = 1, πv1 (b1 ) = 2 and
πv1 (b2 ) = 0, since π is balanced. This labelling is shown in Figure 3.12 (a). So all star
labellings consistent with W have the same labels, and thus the same sign, for V (H) = {v1 }.
For H of Type (2), since π is balanced, the labelling must have labels πv1 (bH ) = 1,
πv2 (b2 ) = 0 and πvi (bi ) = 1 for i = 3, . . . , n. The edge weight of e1 is W (e1 ) = 3 and
πv1 (bH ) = 1, so π must have πv1 (e1 ) = 2 and πv2 (e1 ) = 1. We must have πv1 (en ) = 0, as
πv1 (bH ) = 1 and πv1 (e1 ) = 2. The edges ei for i = 2, . . . n have edge weight 2. These edges
must have labels 2 and 0, since πvi (bi ) = 1 for i 6= 1, 2. In fact, πvi (ei ) = 2 and πv(i+1) (ei ) = 0
(indices are taken modulo n) since they must agree with the label πv1 (en ) = 0. This labelling
is shown in Figure 3.12 (b). So all star labelling consistent with W have the same labels,
and thus the same overall sign, for V (H).
∗
Finally, we consider H of Type (3) with the 2-factor F H chosen in Section 3.2.2. First
we show that for each thread t, there is a unique star labelling for the vertices in Vin (t)
consistent with the edge weights W . This unique star labelling is illustrated in Figure 3.13.
We distinguish certain labels, by “boxing” or “triangling” them in Figure 3.13; these labels
are important in the following argument.
Let et ∈ E(H ∗ ) be the corresponding thread edge to the thread t. We recall the edge
weights W (e) of the edges e ∈ E(t)∪B(t) assigned in Section 3.2.2 and shown in Figure 3.10.
CHAPTER 3. CUBIC GRAPHS
Case(1,2):
1
v1
Case(1,1):
1
v1
3
et
2
et
Case(0,2):
2
0
et
v1
Case(0,1):
1
vm+1
Case(0,0):
0
v1
1
et
0
et
24
v1
2
vm+1
v1
1
vm+1
2
vm+1
0
v1
vm+1
v2
11 0 2
e1 1
e2
2
b2
1
2 02 2
1
2
1
22 0 2
1
2
1
v1 0 v2 3
0 0 1
2
e1 2
e2
3
b2
1
2 02 2
1
2
1
vm+1 3 vm 1
1 2 1
0
em 0
em-1
1
bm
1
2
0 2
0
1
2
1
v1
0
v2
11 0 1 3 2
e1 2
e2
b2 1 3
0
v2 2
0
0 2
0
e1 1
e2
2
b2
1
2
0 2
0
1
2
1
Figure 3.13: The unique star labellings for threads.
vm+1
2
em
3 vm+1
0 2 1
1 em
2
bm
1
vm+1
2
em
e1
v1
0
vm+1
0
em
CHAPTER 3. CUBIC GRAPHS
25
We will show that the labels πv for v ∈ V (t) are as shown in Figure 3.13. We suppose t is
not a root thread. In each case the squared labels, πvi (bi ), are determined to be as shown
by Lemma 3.2.2. Next, we see that the triangled label, πv1 (e1 ), πv1 (e2 ) or πvm (em−1 ) is as
shown, based on the edge weight W (e1 ), W (e2 ) or W (em−1 ) and the fact that each vertex
must be incident to the labels 0, 1 and 2. Finally, we see that the rest of the labels πv (ei )
are as shown, again since π is a star labelling consistent with W .
If t is a root thread, then t has edge weights W given by either Case(1,1) or Case(0,0). In
Figure 3.13, we see that in both these cases all cut edges b ∈ B(t) have the label πup(b) (b) = 1.
If vj is the root vertex of H, then πvj (bH ) = 1, and the remaining labels of π are as above.
Thus, there is a unique star labelling that is consistent with the edge weighting W for the
vertices Vin (t) of the root thread of H.
For each star labelling π of the vertices V (H) that is consistent with W , we can define
a star labelling π 0 of H ∗ by πv0 = πv for v ∈ V (H ∗ ) (where we identify πv0 1 (et ) = πv1 (e1 ) and
πv0 m+1 (et ) = πvm+1 (em ) for each thread t = v1 e1 . . . em vm+1 and corresponding thread edge
et = v1 vm+1 . The star labelling π 0 is consistent with W H
Figure 3.13 πv1 (e1 )+πvm+1 (em ) =
∗
H
W H ,F
∗ ,F H ∗
∗
(et ), and W (e) =
since for all labellings in
∗
H
W H ,F
∗
(e) for all non-thread
edges e. By Lemma 3.1.1 and Lemma 3.1.3, sgn(π 0 ) is the same for all choices of π.
For every star labelling π consistent with W , πv is as shown in Figure 3.13 for v ∈ V (t)
and every thread t of H so,
Y
sgn(πv ) =
Y
sgn(πv ) ·
Y
{sgn(πv ) : v ∈ Vin (t) for a thread t of H}
sgn(πv0 ) ·
Y
{sgn(πv ) : v ∈ Vin (t) for a thread t of H}
v∈V (H ∗ )
v∈V (H)
=
Y
v∈V (H ∗ )
= sgn(π 0 ) ·
Y
{sgn(πv ) : v ∈ Vin (t) for a thread t of H}
and so the signs of all star labellings π that are consistent with W are the same.
3.2.4
Proof of Theorem 1.0.3
Proof of Theorem 1.0.3 We can construct a star labelling π of G consistent with W
as follows. We let πlow(b) (b) = 1 for every b ∈ B. For vertices of H ∈ Comp(G \ B) of
Type (1) or Type (2), we let πv be as shown in Figure 3.12. For each H ∈ Comp(G \ B)
∗
of Type (3), we choose a 2-factor F H as in in Section 3.2.2 and we fix a star labelling π ∗
of H ∗ consistent with W H
∗ ,F H ∗
; such a star labelling is illustrated by π1 in Section 3.1.1
CHAPTER 3. CUBIC GRAPHS
26
or by ρ in Section 3.1.2. Then we let πv = πv∗ for all v ∈ V ∩ V (H ∗ ). Finally, for each for
v ∈ Vin (t) for every thread t of H, we let the star labelling πv be as shown in Figure 3.13,
where the cases are defined as in Section 3.2.2 and depend on π ∗ above.
By Lemma 3.2.3, all star labellings of G that are consistent with the edge weighting W ,
as defined in Section 3.2.2, have the same sign. So the number of positive and negative star
labellings must be different, and by Theorem 2.0.6, G is 4-edge-choosable since W ≤ 3.
Chapter 4
Cubic Planar Graphs
In this chapter, we consider any connected cubic planar graph G and prove Theorem 1.0.4.
4.1
2-Edge-Connected Cubic Planar Graphs
We consider a 2-edge-connected cubic planar graph G. By a consequence of the Four Colour
Theorem [13], G is 3-edge-colourable. Then, by [6, Theorem 1.2] of Ellingham and Goddyn,
G is 3-edge-choosable. Their result is actually more general; it asserts that k-edge-colourable
k-regular planar graphs are k-edge-choosable.
For Section 4.2 we need another theorem of [6, p.345 and Theorem 3.1] regarding the
star labellings of G. First, we must define the following terminology. We let 2 be the edge
weighting of a graph that assigns each edge weight 2. For a star labelling π of any graph,
we consider the subgraph with edges {e = uv : πu (e) = 0, πv (e) = 2}. The cycles of this
subgraph are called 0-2 cycles of the graph, and π is 0-2 bipartite if all the 0-2 cycles
have even length.
Theorem 4.1.1 If G is a 2-edge-connected cubic planar graph, then all 0-2 bipartite star
labellings π of G consistent with 2 have the same sign.
27
CHAPTER 4. CUBIC PLANAR GRAPHS
4.2
4.2.1
28
General Cubic Planar Graphs
Structure of the Graph
We consider a 2-edge-connected cubic planar graph G. Many of the definitions in this section
are the same as in Section 3.2.1, but we repeat some here for clarity. In particular, we let
B ⊂ E be the set of cut edges of G, so G\B consists of 2-edge-connected components and
isolated vertices. As in Section 3.2.1, each H ∈ Comp(G \ B) must be of one of the following
types:
Type (1) a single vertex,
Type (2) a cycle, or
Type (3) a graph with at least two vertices of degree 3.
Again, we let T be the tree with vertices H ∈ Comp(G \ B) and edges B, and we root T
at a leaf Hr . For each H ∈ Comp(G \ B), the root vertex and the bridge edges below H are
as before. For H is of Type (3), we recall the definition of a thread t = v1 e1 . . . vm em vm+1
of H. Again, we construct the graphs H ∗ with corresponding thread edges et = v1 vm+1 ,
including the root thread edge. We recall the definition of
G∗ = ∪ {H ∗ : H ∈ Comp(G \ B), H is of Type (3)}.
As in Section 3.2.1, we will distinguish certain vertices and edges of H ∈ Comp(G \ B)
and B(H). Recall that for H of Type (1), H is the vertex v1 , and the two cut edges in B(H)
are b1 and b2 . If H is of Type (2) with root vertex v1 , H is the cycle v1 e1 v2 e2 . . . vn en v1 ,
and the edges in B(H) are bi = B(vi ) for i = 2, . . . , n. If H is of Type (3), then each thread
t of H is t = v1 e1 . . . vm em vm+1 and we let bi = B(vi ) for i = 2, . . . m. If vj ∈ V (t) is the
root vertex of H then bj is not defined, rather vj is adjacent to the cut edge bH above it.
Every component H ∗ of G∗ is 3-regular, planar and 2-edge-connected, and thus 3-edgecolourable by the Four Colour Theorem. We fix a 3-edge-colouring φ → {red, green, blue}
of G∗ such that:
(A) each root edge is coloured blue, and
(B) the number of blue thread edges is maximal subject to (A).
Such a edge-colouring exists since each root edge is in a distinct component of G∗ .
(4.1)
CHAPTER 4. CUBIC PLANAR GRAPHS
4.2.2
29
Edge Weightings of the Graph
The following procedure assigns a primary edge weight to each edge in E, and an additional
secondary edge weight to each edge of certain threads of G. For an edge e, we denote the
primary edge weight by wp (e) and, if it exists, we denote the secondary edge weight by
ws (e). Then we define a set of edge weightings W of G; an edge weighting will assign either
the primary or secondary edge weight at each edge.
We consider each H ∈ Comp(G \ B) in turn and assign a primary weight, and possibly a
secondary weight, to each edge in E(H) ∪ B(H). If H is of Type (1) or Type (2), then only
a primary weight is assigned to an e ∈ E(H) ∪ B(H). If H is of Type (3), then we assign
a primary edge weight to each edge e ∈ E(H) ∪ B(H), and we may also assign a secondary
edge weight to a subset of these edges.
The edge weights for H of Type (1) and Type (2) are as in Section 3.2.2. In particular,
if H is of Type (1), then the primary edge weights are wp (b1 ) = 3 and wp (b2 ) = 1. Again
as in Section 3.2.2, if H is of Type (2), then the primary edge weights are wp (e1 ) = 3,
wp (b2 ) = 1, and wp (f ) = 2 for f ∈ (E(H) ∪ B(H)) \ {e1 , b2 }. These edge weights are shown
in Figure 3.9.
Finally, we consider H of Type (3). First, we define a partial orientation on the corresponding H ∗ . The red and green edges of H ∗ under φ induce cycles, which we orient
clockwise. We assign only the primary edge weight wp (e) = 2 for each non-thread edge e of
H. We consider each thread t of H separately. For each thread, we assign edge weights to
the edges in E(t) ∪ B(t). Let et be the thread edge of H ∗ corresponding to t. We proceed
as follows.
In the case where φ(et ) = blue, we assign only a primary edge weight to each edge
in E(t) ∪ B(t). Note the et may be the root edge of H ∗ in this case. These weights are
wp (e1 ) = 1, wp (em ) = 3 and wp (f ) = 2 for every other edge f ∈ (E(t) ∪ B(t)) \ {e1 , em }.
In the case where φ(et ) ∈ {red, green}, we assign both a primary edge weight and a
secondary edge weight for each edge in E(t) ∪ B(t). Note that et = v1 vm+1 cannot be the
root edge of Hi∗ and that et has an orientation; we may assume the orientation is from v1 to
vm+1 . We assign the primary edge weights of wp (em ) = 3, wp (em−1 ) = 0, wp (bm ) = 3, and
wp (f ) = 2 for f ∈ (E(t) ∪ B(t)) \ {em−1 , em , bm } . The secondary edge weights depend on
the parity of m. If m is odd, then we assign the secondary weights ws (e1 ) = 3, ws (e2 ) = 0,
ws (e2 ) = 3, and ws (f ) = 2 for f ∈ (E(t) ∪ B(t)) \ {e1 , e2 , b2 }. Finally, if m is even,
CHAPTER 4. CUBIC PLANAR GRAPHS
30
Type (3):
vm+1
blue:
primary:
red/green:
primary:
em 2
v1
v1
secondary odd:
v1
secondary even:
3
2
e1
2
v2
2
2
b2
v2
3
0
e1 3
b2
2
e1
2
2
v2
v1
vm
vm+1
1
2 e1
0
2
2
2
3
em
3
bm
vm+1
2
em
3
vm
vm+1
2
em
1
bm
Figure 4.1: The primary and secondary edge weights for the edges of threads of
H ∈ Comp(G \ B) of Type (3).
then we assign secondary edge weights ws (em−1 ) = 3, ws (bm ) = 1,and ws (f ) = 2 for
f ∈ (E(t) ∪ B(t)) \ {em−1 , bm }. The primary and secondary edge weights are illustrated in
Figure 4.1.
We now define the edge weightings of the graph G. Let R be the set of threads t of G such
that φ(et ) ∈ {red, green}. The set of edge weightings W is defined to be W = {WS : S ⊂ R},
where for each S ⊂ R the edge weighting WS ∈ {0, 1, 2, 3}E(G) is

ws (e) if e ∈ S
t∈S (E(t) ∪ B(t))
WS (e) =
w (e) otherwise.
p
We note that all edge weightings WS of G have average edge weight 2 over E(H) ∪ B(H)
for H ∈ Comp(G \ B).
CHAPTER 4. CUBIC PLANAR GRAPHS
31
2
2
3
H6
2
1
2
2
2
2
H5
3
2
1
2
2
2
H4
1
2
3
3
2
1
3
3
3
1
H10
2
H2
H3
1
2
1
2
2
2
2
2
2
2
1
2
3
1
2
3
1
2
2 3
H7
2
H1
2
2
2
H8
2
2
H9
3
2
1
2
2
Figure 4.2: The edge weighting W∅ of the example graph from Figure 3.8.
CHAPTER 4. CUBIC PLANAR GRAPHS
4.2.3
32
Reference Star Labelling of the Graph
In this section, we define a reference star labelling of G, ρ, that is balanced, 0-2 bipartite,
and consistent with the edge weighting W∅ . Recall that a star labelling π is balanced if for
every cut edge b, πlow(b) (b) = 1. The labels of ρ are assigned by considering the vertices
V (H) for each H ∈ Comp(G \ B) in turn. Since ρ is balanced, we know that ρ`(b) (b) = 1
for all cut edges b ∈ B.
For H ∈ Comp(G \ B) of Type (1), the labelling is ρv (b1 ) = 2 and ρv (b2 ) = 0. Since ρ is
balanced, ρ is consistent with W∅ on δ(v), and v is incident to the distinct labels 0, 1, and
2. The star labelling in shown Figure 4.3.
For H ∈ Comp(G \ B) of Type(2), the labelling is ρv1 (e1 ) = 2, ρv1 (en ) = 0, ρv2 (e1 ) = 1,
ρv2 (b2 ) = 0, ρv2 (e3 ) = 2, and for i = 3, . . . , n, ρvi (ei−1 ) = 0, ρvi (ei ) = 2 and ρvi (bi ) = 1. By
construction, ρ is consistent with W∅ for the edges in E(H), and ρ is consistent with W∅ for
the edges in B(H), since ρ is balanced. Again, each vertex is incident to distinct labels 0,
1 and 2, and the star labelling in shown Figure 4.3.
To help define π for H of Type (3), we define a star labelling of G∗ , ρ∗ , based on φ and
the partial orientation from Section 4.2.2 so that ρ∗ is consistent with the edge weighting 2
of G∗ . For every blue edge e = uv of G∗ , the labelling is ρ∗u (e) = 1 and ρ∗v (e) = 1. Every
red or green edge e = uv has an orientation, say from u to v. Based on the orientation, we
define the labels ρ∗u (e) = 0 and ρ∗v (e) = 2. Since φ is a proper 3-edge-colouring, every vertex
in V (G∗ ) is incident to the labels 0, 1 and 2, and ρ∗ is 0-2 bipartite.
Finally, we use ρ∗ to define ρ for H ∈ Comp(G \ B) of Type (3). For v ∈ V (H ∗ ), we
assign labels ρv = ρ∗v (for each thread t = v1 e1 . . . em vm+1 and corresponding thread edge
et = v1 vm+1 , we identify πv0 1 (et ) = πv1 (e1 ) and πv0 m+1 (et ) = πvm+1 (em )). The vertices in
S
V (H) \ V (H ∗ ) are the internal vertices of the threads of H, {Vin (t) : t a thread of H}. For
each thread t = v1 e1 . . . em vm+1 of H, the vertices Vin (t) are labelled based on the colour
φ(et ) as follows:
In the case where φ(et ) = blue, the labels ρv1 (e1 ) = 1 and ρvm+1 (em ) = 1 are assigned
above. The internal vertices have the labels ρv2 (e1 ) = 0, ρvm (em ) = 2, and for i = 2, . . . m−1,
ρvi (ei ) = 2 and ρvi+1 (ei ) = 0. The labelling at each bi ∈ B(t) is ρvi (bi ) = 1. We recall that
the edge weights are W∅ (e1 ) = 3, W∅ (em ) = 1, and for i = 2, . . . m − 1, W∅ (ei ) = 2 and
W∅ (bi ) = 2. As illustrated in Figure 4.3, the star labellings {ρv : v ∈ Vin (t)} are consistent
with W∅ for the edges E(H) ∪ B(H).
CHAPTER 4. CUBIC PLANAR GRAPHS
3
2
2
2
2
v2
e1 1
2
0
2 1
2
0
1
0 b2
2
1
1
0
2
1
2
2 0
b1 b2
v1 1
1
3
v1 1
1
Type (2):
Type (1):
1
33
1
Type (3):
blue:
vm+1
3
1 2 10
em
2
1
2
2
2
red/green:
10
1
2
2
2
2
1
0 1
1 e1
1
v1
v1
0 e1
2
2
2
10
1
Figure 4.3: The star labelling ρ for H ∈ Comp(G \ B).
0
0 0
3
3
1 2
2 em
bm
1
vm+1
CHAPTER 4. CUBIC PLANAR GRAPHS
34
In the case where φ(et ) ∈ {red, green}, the labels ρv1 (e1 ) = 0 and ρvm+1 (em ) = 2 are
assigned above. The reference labelling is ρvm (em ) = 1, ρvm (em−1 ) = 0, ρvm−1 (em−1 )=0,
ρvm (bm ) = 2, ρvi (bi ) = 1 for i 6= m−1, and for i = 1, . . . , m−2, ρvi (ei ) = 0 and ρvi (ei−1 ) = 2.
We recall that the edge weights are W∅ (em−1 ) = 0, W∅ (em ) = 3 and W∅ (bm ) = 3, and all
other edges in E(t)∪B(t) have edge weight 2. As illustrated in Figure 4.3, the star labellings
{ρv : v ∈ Vin (t)} are consistent with W∅ for the edges E(H) ∪ B(H).
We have shown that the reference labelling ρ is valid star labelling at every vertex of G.
By construction and since ρ is balanced, this reference labelling is consistent with W∅ . We
can see that every 0-2 cycle in G under ρ is a 0-2 cycles of G∗ under ρ∗ , and so each has
even length and thus ρ is 0-2 bipartite. We have shown Lemma 4.2.1.
Lemma 4.2.1 The reference star labelling for a planar cubic graph G, ρ, as defined above
is consistent with W∅ and is 0-2 bipartite. The reference star labelling ρ∗ of G∗ , as above,
is consistent with 2 and 0-2 bipartite.
4.2.4
Arbitrary Star Labellings of the Graph
In this section, we consider any arbitrary star labelling π of G that is consistent with any
of the edge weightings WS ∈ W. We let the set of all such star labellings be
Π = {π : π is a star labelling of G consistent with some WS ∈ W}.
Lemma 4.2.2 Every star labelling in Π is balanced.
Proof The proof is the same as that of Lemma 3.2.2 in Chapter 3, recalling from the end of
Section 4.2.2 that the average edge weight over E(H) ∪ B(H) is 2 for all H ∈ Comp(G \ B).
The next lemma shows that if π is a star labelling consistent with some WS ∈ W, then
for certain v ∈ V (G), πv is one of only several possible labellings.
Lemma 4.2.3 We consider WS ∈ W and π ∈ Π that is consistent with WS . Let H ∈
Comp(G \ B). If H is of Type (1) or Type (2), then the star labelling must be πv = ρv for
every vertex v ∈ V (H).
If H is of Type (3), then for each thread t = v1 . . . vm+1 6∈ S of H and vertex v ∈ Vin (t),
the star labelling must be πv = ρv . For each thread t = v1 . . . vm+1 ∈ S of H if m > 2, then
CHAPTER 4. CUBIC PLANAR GRAPHS
35
H2
0
2 1
H1
2
0
3
H
2 6
2
0
1
2 2
0 21
0
2
1
3
1
1
0
2
1
0
0
21
2
1
2
2
2
2
3
2
1
2 0
1
2
0
2
2
H5
2
2
1
1
2
0
2
2
0
3
2
H4
1
2
1
0
3
2
1
2
2 2
0
0
1
3
2
0
11
H10
2
2
1
1
0
1
02
2 1
2
0
3
1
1 0 2
1
0
2
0
1
2
2
0
0
2
0
2
H8
2
2
2
H9 01
1 2
12
3
1
2
2
H3
1
3
2
1
2
1
0
1
1
0
2
1
2 0
1
2
H7
1
2 0
3
1
2
0
1
2
2
1
2
2
1
0 22
0
0
2 1
2 2
1
2 2 1
0
0
3
1
2 2
2
2
0 1
2
0
Figure 4.4: The reference star labelling ρ for the example graph from Figure 3.8.
CHAPTER 4. CUBIC PLANAR GRAPHS
36
the star labelling πvi for i = 2, . . . , m is as illustrated in Figure 4.6 in the secondary odd
case or secondary even (option 1) case, depending on the parity of m. If m = 2, then the
star labelling πvi for i = 2, . . . , m is as illustrated in Figure 4.6 in either the secondary even
(option 1) case or secondary even (option 2) case.
Proof We consider some π ∈ Π which is consistent with an edge weighting WS ∈ W. It
follows from Lemma 4.2.2 that πlow(b) (b) = 1 and πup(b) (b) = WS (b) − 1 = ρup(b) (b) for every
cut edge b ∈ B. First, we consider H ∈ Comp(G \ B) of Type (1), so V (H) = {v1 } and
δ(v1 ) = {bH , b1 , b2 } ⊂ B. Since all edge weightings in W assign the same edge weights to
{b1 , b2 }, πv1 = ρv1 by Lemma 4.2.2.
Next, we consider π on the vertices of H ∈ Comp(G \ B) of Type (2). Again, all edge
weightings in W assign the same edge weights for E(H) ∪ B(H). By Lemma 4.2.2, we know
that πv2 (b2 ) = ρv2 (b2 ) = 0, and πvi (bi ) = ρvi (bi ) = 1 for i = 3, . . . , n. Since WS (e1 ) = 3, we
must have {πv1 (e1 ), πv2 (e1 )} = {1, 2}. Because π is balanced, we know the label πv1 (bH ) = 1,
where bH is the cut edge above v1 . Thus, the labels must be πv1 (e1 ) = 2, πv2 (e1 ) = 1
and πv1 (en ) = 0, since v1 must be adjacent to the labels 0, 1 and 2. The edges ei , for
i = 2, . . . n, have edge weight Ws (ei ) = 2 and must have labels {πvi (ei ), πvi+1 (ei )} = {0, 2},
since πvi (bi ) = 1 for i 6= 1, 2 (indices are taken modulo n). These labels must be the same
for each i and πv1 (en ) = 0, so the labels are πvi (ei ) = 2 and πvi+1 (ei ) = 0. Thus we see that
πu = ρu for u ∈ V (H).
Finally, we consider π for the vertices of threads of H ∈ Comp(G \ B) of Type (3). We
let t be a thread and et be the corresponding thread edge of H ∗ . If φ(et ) = blue and et is not
the root thread edge, then all edge weighting WS have the same edge weights for E(t)∪B(t),
as illustrated in Figure 4.1. Since π is balanced, we must have the labels πvi (bi ) = ρvi (bi )
for i = 1, . . . m, which are shown as the boxed labels in Figure 4.5. Next, since W (em ) = 3,
the triangled label must be πvm (em ) = 2 = ρvm (em ). By WS and since every vertex must
be incident to labels 0, 1 and 2, the rest of labels are as shown in Figure 4.5. In the case
where et is the root thread edge, the root vertex vj ∈ V (t) is incident with bH and B(vj )
does not exist. Since π is balanced, we still have πvj (bH ) = 1 and so the labels in V (t) are
as in the non-root case. So for all threads with φ(et ) = blue, the star labelling is πvi = ρvi
for all vertices vi ∈ V (t).
Now we consider the case where φ(et ) ∈ {red, green}. We will show that the star
labellings are as shown in Figure 4.6. In the figure, we again distinguish certain labels, by
CHAPTER 4. CUBIC PLANAR GRAPHS
1
3
em
2
1
1
2
0
2
2
1
1
37
0
2
2
2
1
1
1
0
e1
2
1
Figure 4.5: The general star labelling π for a blue thread of H ∈ Comp(G \ B) of Type(3).
“boxing” or “triangling” them. These labels are important in the following argument.
We recall that if t 6∈ S, then WS assigns the edges in E(t) ∪ B(t) primary edge weights,
but if t ∈ S, then WS assigns the edges in E(t) ∪ B(t) have secondary edge weights. Both
of these sets of edge weights are illustrated in Figure 4.1. We first suppose that t 6∈ S (resp.
that t ∈ S and m is odd). Since π is balanced, the labels of up(b) for b ∈ B(t) are as
shown by the boxed labels of Figure 4.6. Since the edge weighting gives WS (e2 ) = 0 (resp.
WS (em−1 ) = 0), the triangled label must be πv1 (e2 ) = 0 (resp. πvm (em−1 ) = 0). By the
edge weights and since every vertex must be incident to the labels 0, 1 and 2, the remaining
labels are in the primary (resp. the secondary odd) illustrations of Figure 4.6.
Finally if t ∈ S and m is even, then there are two possibilities for the labels assigned
by π. Again, π is balanced and the labels πup(b) (b) for b ∈ B(t) are as shown by the boxed
labels in Figure 4.6. Next, the triangled label can be either πv2 (e2 ) = 2 or πv2 (e2 ) = 1. If
the triangled label is πv2 (e2 ) = 1, then the rest of the labels must be as shown in Figure 4.6
secondary even (option 1) illustration. If the triangled label is πv2 (e2 ) = 2, then t must
have had length m = 2, as otherwise WS would force the labels πv3 (b3 ) = 1 = πv3 (e2 ) as in
Figure 4.7 and πv3 would not be a star labelling. The labels when m = 2 are as shown in
CHAPTER 4. CUBIC PLANAR GRAPHS
primary:
2
3
e1
1
0
0
0
0
2
1
1
1
b2 3
secondary odd:
0
2
e1
0
2
1
e1
2
1
2
2
em
0
2
0
0
2
0
0
1
2
bm 3
2
1
secondary even:
38
3
em
2
1
3
2
2
1
0
b2 1
1
1
1
2
e1
0
2
0
2
em
2
(option 1)
2
3
1
2 1
0
e2
(option 2)
b2 1
1
Figure 4.6: The general star labelling π for red or green threads of H ∈ Comp(G \ B) of
Type (3).
Figure 4.6 secondary even (option 2) illustration.
We formally name the sets of star labelling of the interior vertices of threads t of G
seen in the proof above, as we will need to refer to them. They are the primary case,
the secondary odd case, the secondary even (option 1) case and the secondary
even (option2) case star labellings of a thread t, and are defined to be as illustrated in
Figure 4.6.
CHAPTER 4. CUBIC PLANAR GRAPHS
1
2
e1
1
39
3
2
2 1
0
b2 1
1
1
1
2
impossible!
Figure 4.7: Applying secondary even (option 2) case labels to a thread of length m ≥ 2
leads to a contradiction.
4.2.5
Signs of the Star Labellings of the Graph
For each star labelling π of G we define a star labelling π ∗ of
G∗ = ∪{H ∗ : H ∈ Comp(G\B), H is of Type (3)}. Note that the vertices of G∗ are a subset
of the vertices of G, while the edges of G∗ can be decomposed into the sets E1∗ = E(G∗ ) ∩ E
and E2∗ = {et : t is a thread of G} so that E(G∗ ) = E1∗ ∪ E2∗ . The star labelling π ∗ is defined
as follows.
πv∗ (e) =
πv∗1 (et ) =
πv (e)
πv1 (e1 )
πv∗m+1 (et ) = πvm+1 (em )
for e ∈ E1∗ with endpoint v
for et ∈ E2∗ with corresponding t = v1 e1 . . . em vm+1
for et ∈ E2∗ with corresponding t = v1 e1 . . . em vm+1
Every vertex v ∈ V (G∗ ) retains the labels 0, 1 and 2, so π ∗ is a star labelling of G∗ . We
see that if π = ρ, the reference star labelling defined in Section 4.2.3, then π ∗ = ρ∗ . We let
ρ∗ be the reference labelling of G∗ and we see that sgn(πv ) = sgn(πv∗ ) for v ∈ V (G∗ ), since
the order of the labels is the same at every vertex (we identify each thread edge et with the
edge e1 at v1 and the edge em at vm + 1). The edges e ∈ E1∗ have edge weight WS (e) = 2
for all WS ∈ W. For all threads t of G, any π ∈ Π has πv1 (e1 ) + πvm+1 (em ) = 2, as we see
in Figure 4.5 and Figure 4.6 . Thus, π ∗ is consistent with the edge weighting 2 of G∗ .
CHAPTER 4. CUBIC PLANAR GRAPHS
40
Lemma 4.2.4 If π ∈ Π is a star labelling of G that is consistent with some edge weighting
WS ∈ W, then sgn(π) = sgn(π ∗ )·(−1)n , where n is the number of threads t of length 2 whose
vertices have the labels given by the secondary even (option 2) case labelling (illustrated in
Figure 4.6).
Proof We calculate the sign of π by comparing it to the reference star labelling ρ. By
definition, the sign of a star labelling π of G, is the product of the signs of the star labellings
of its vertices. We define the set of vertices
V 0 =V \ V (G∗ )
[
= {V (H) : H ∈ Comp(G \ B), H is of Type (1) or Type (2)}
[
{Vin (t) : t is a thread of G}
So then,
sgn(π) =
Y
sgn(πv )
v∈V (G)

= 

Y
!
sgn(πv ) ·
sgn(πv )
v∈V 0
v∈V (G∗ )

= 
Y

Y
v∈V
sgn(πv∗ )
(G∗ )
!
Y
·
v∈V
sgn(πv )
0
!
∗
= sgn(π ) ·
Y
v∈V
sgn(πv )
0
We consider the first factor, sgn(π ∗ ), of this product. Since G∗ is planar, 3-regular and
3-edge-colourable, all star labellings π ∗ of G∗ that are consistent with 2 and 0-2 bipartite
have the same sign by Lemma 4.1.1. Their sign is positive since the reference labelling ρ∗
is one such labelling, by Lemma 4.2.1. If π ∗ has an odd 0-2 cycle, then sgn(π ∗ ) may be
positive or negative.
Y
We calculate
sgn(πv ) directly. By Lemma 4.2.3, a star labelling π of G that is
v∈V 0
consistent with any WS has πv = ρv for vertices v of H ∈ Comp(G \ B) of Type(1) and
Type(2), and in Vin (t) for each thread t of G, such that φ(et ) = blue. Thus, sgn(πv ) is
positive for these vertices.
CHAPTER 4. CUBIC PLANAR GRAPHS
41
Again by Lemma 4.2.3, there are several possible star labellings of the vertices in
Vin (t) for a thread t = v1 e1 . . . em vm+1 of G where the corresponding thread edge et has
φ(et ) ∈ {red, green}. These labellings are illustrated in Figure 4.6. We calculate the
Y
overall sign
sgn(πv ) of the internal vertices for each such thread. First, if t is of
v∈Vin (t)
odd length, then these vertices have labels from either the primary case star labellings or
secondary odd case star labellings as shown in Figure 4.6. The primary case star labelling
is πv = ρv for all v ∈ Vin (t). So each πv has positive sign, and the overall sign for the
vertices Vin (t) is positive. If the vertices in Vin (t) have the secondary odd case star labels,
then the labels are πvi (bi ) = ρvi (bi ), πvi (ei ) = ρvi (ei−1 ) and πvi (ei−1 ) = ρvi (ei ), for each
vertex vi ∈ Vin (t), i = 3, . . . , m − 1. The labels of v2 are πv2 (b2 ) = ρv2 (e1 ), πv2 (e2 ) = ρv2 (e2 )
and πv2 (e1 ) = ρv2 (b2 ), and the labels of vm are πvm (bm ) = ρvm (em ), πvm (em ) = ρvm (bm )
and πvm (em−1 ) = ρvm (em−1 ). So each πvi differs by a single transposition from ρvi and
sgn(πvi ) is negative. Since there are an even number of internal vertices, the overall sign of
the internal vertices of t is positive.
Next, we consider an even thread t with φ(et ) ∈ {red, green}, which has three possible
labellings for Vin (t): the primary case labellings, the secondary even (option 1) case labellings
or the secondary even (option 2) case labellings. The primary case labelling is πv = ρv for
all v ∈ Vin (t). So each πv has positive sign and the overall sign for the vertices Vin (t) is
positive. If π assigns the secondary even (option 1) case labels to the vertices in Vin (t), then
the labels for i = 3, . . . , m are πvi (bi ) = ρvi (bi ), πvi (ei ) = ρvi (ei−1 ) and πvi (ei−1 ) = ρvi (ei ).
So for i = 3, . . . , m, the star labelling πvi differs by a single transposition from ρvi and
has negative sign. The remaining vertex v2 in Vin (t) has the labelling πv2 (b2 ) = ρv2 (e2 ),
πv2 (e2 ) = ρv2 (e1 ) and πv2 (e1 ) = ρv2 (b2 ), and so πvm differs by two transpositions from ρvm
and has positive sign. Since there are an even number of vertices of Vin (t) of negative sign
and one vertex of positive sign, the overall sign of the internal vertices of t is positive.
Finally, if π assigns the secondary even (option 2) case labels to the vertices in Vin (t),
then t has length 2 and Vin (t) = {e2 }. The labels are πv2 (b2 ) = ρv2 (e2 ), πv2 (e2 ) = ρv2 (b2 )
and πv2 (e1 ) = ρv2 (e1 ). So πv2 differs by a single transposition from ρv2 and so sgn(πv2 ) is
negative (also the overall sign of Vin (t) ).
Thus, for every star labelling π that is consistent with some weighting WS ∈ W and
that assigns secondary even (option 2) case star labellings to exactly n threads of length 2,
Y
the overall sign of π on the vertices in V 0 is
sgn(πv ) = (−1)n . Thus the sign of star
v∈V 0
CHAPTER 4. CUBIC PLANAR GRAPHS
42
!
∗
labelling π of G is sgn(π) = sgn(π ) ·
Y
v∈V
4.2.6
sgn(πv ) .
0
Proof of Theorem 1.0.4
We let Π0 ⊂ Π be the set of star labellings π for which π ∗ is 0-2 bipartite. Then π is also
0-2 bipartite, since labellings of the threads or of H ∈ Comp(G \ B) of Type (1) or Type (2)
cannot have labels that complete a 0-2 cycle and are consistent with some WS ∈ W, as we
see in Figure 4.3, Figure 4.5 and Figure 4.6. We know that ρ ∈ Π0 by Lemma 4.2.1. We
let Π1 ⊂ Π be the star labellings π of G such that π ∗ is not 0-2 bipartite. Note that π may
still be 0-2 bipartite if each odd 0-2 cycle of π ∗ contains a thread edge. We have partitioned
Π = Π0 ∪ Π1 .
Lemma 4.2.5 For π ∈ Π0 , sgn(π) is positive.
Proof Consider a star labelling π ∈ Π0 of G, which is consistent with WS ∈ W. Then π ∗
is 0-2 bipartite and induces a 3-edge-colouring ψ of G∗ by colouring the edges of 0-2 cycles
alternating red and green, and colouring the edges labelled 1 on both ends blue.
We consider a thread t of G and the corresponding thread edge et of G∗ . If φ(et ) = blue,
then the edge weighting WS assigns primary edge weights for e ∈ E(t) and so π assigns label
as shown in Figure 4.5, in particular πv1 (e1 ) = 1 = πvm+1 (em ). So πv∗1 (et ) = 1 = πv∗m+1 (et )
and thus ψ(et ) = blue. Then {et : φ(et ) = blue} is a subset of {et : ψ(et ) = blue}. But
then, by the maximality condition (4.1), these sets must be equal:
{et : φ(et ) = blue} = {et : ψ(et ) = blue}.
Now, we suppose that φ(et ) is either red or green. If t has length greater than 2, then π
assigns labels πv1 (e1 ) = 0 and πvm+1 (em ) = 2, or πvm+1 (em ) = 0 and πv1 (e1 ) = 2 as seen in
Figure 4.6 in the primary case, secondary odd case and secondary even (option 1) case star
labellings. Thus, ψ(et ) is also red or green. If t has length 2 and π assigns the vertices in
Vin (t) the primary case labelling or the secondary even (option 1) case labelling, then ψ(et )
is either red or green again. However, if π assigns the vertex in Vin (t) the secondary even
(option 2) case labelling, then πv1 (e1 ) = 1 = πv3 (e2 ) and ψ(et ) = blue, a contradiction since
φ(et ) ∈ {red, green} but {et : φ(et ) = blue} = {et : ψ(et ) = blue} from above.
Thus, if π ∗ is 0-2 bipartite, then π cannot assign any thread labels from the secondary
even (option 2) case. That is, n = 0 in Lemma 4.2.4. We recall that, since π ∗ is
CHAPTER 4. CUBIC PLANAR GRAPHS
43
0-2 bipartite, it has positive sign by Theorem 4.1.1. Thus by Lemma 4.2.4, the sign is
sgn(π) = (+1)(−1)0 = (+1), which is positive.
Lemma 4.2.6 The set of star labellings Π1 contains an equal number of positive and negative star labellings.
Proof We prove this lemma by defining a sign reversing involution Ω on the star labellings
of Π1 . We fix an ordering of the odd cycles of G∗ , and we consider a star labelling π ∈ Π1 ,
which is consistent with WS ∈ W. The graph G∗ has at least one odd 0-2 cycle under π ∗ ,
and we designate C = u1 f1 u2 . . . uk fk u1 to be the first odd 0-2 cycle.
First, we consider the special case where C is also a cycle of G. Then Ω(π) is the star
labelling of G that interchanges the labels 0 and 2 along C. Formally,
Ω(π)ui (fi ) = πui (fi−1 )
Ω(π)ui (fi−1 ) =
Ω(π)v (e) =
πui (fi )
πv (e)
for i = 1, . . . k
for i = 1, . . . k
for all other incident pairs (v, e) of G
where the indices are taken modulo k.
The star labelling Ω(π) is consistent with WS and C is an odd 0-2 cycle of Ω(π), so
Ω(π) ∈ Π1 . Since π differs from Ω(π) by a transposition at every vertex of C, π and Ω(π)
have opposite signs. The odd 0-2 cycles of G under Ω(π) are the same as those under π,
and so Ω(Ω(π)) = π and Ω is a sign reversing involution for this special case.
Next we define Ω(π) in the case where C is not a cycle of G, and so C must contain at
least one thread edge. We let C 0 be the corresponding cycle of G with all the thread edges
in C replaced by the corresponding threads. We let TC 0 be the set of threads contained in
C 0 , and SC0 ⊂ TC 0 be the set of threads in C 0 whose corresponding thread edges are coloured
red or green under φ. The star labelling Ω(π) is consistent with the edge weighting WS∆SC0 ,
where X∆Y = (X \ Y ) ∪ (Y \ X) is the symmetric difference of sets X and Y . The star
CHAPTER 4. CUBIC PLANAR GRAPHS
44
labelling Ω(π) is defined by the following procedure.
for v ∈ V (G) \ V (C 0 )
(4.2)
πui (fi−1 )
for i = 1, . . . k
(4.3)
πui (fi )
for i = 1, . . . k
(4.4)
Ω(π)v =
πv
For the vertices of V (C) we let:
Ω(π)ui (fi ) =
Ω(π)ui (fi−1 ) =
Ω(π)v (e) =
πv (e)
for all other incident pairs (v, e), v ∈ V (C), (4.5)
where indices are taken modulo k. We have not yet defined Ω(π)v for the vertices of the
threads in TC 0 .
The cycle C is a 0-2 cycle under π ∗ so the labels of π ∗ at the ends of the thread edges
{et : t ∈ TC 0 } are 0 and 2. Thus these thread edges must be coloured red or green under
φ, and the threads have either the primary case, secondary odd case or secondary even
S
(option 1) case labellings. We define Ω(π)v for the remaining vertices t∈T 0 Vin (t). We
C
consider a thread t = v1 e1 . . . em vm+1 ∈ TC 0 , but t 6∈ S and so WS assigns E(t) ∪ B(t)
primary edge weights. Then π assigns Vin (t) the primary case labels and the terminal
vertices the labels πv1 (e1 ) = 2 and πvm+1 (em ) = 0. From (4.3) and (4.4), Ω(π) assigns the
terminal vertices the labels Ω(π)v1 (e1 ) = 0 and Ω(π)vm+1 (em ) = 2. We define Ω(π) to assign
the secondary odd case labels or secondary even (option 1) case labels to the vertices in
Vin (t). These labels are consistent with the secondary edge weights assigned by WS∆SC0 .
Lastly, if t ∈ S and so WS assigns E(t) ∪ B(t) secondary edge weights, then π assigns Vin (t)
the secondary odd case labels or secondary even (option 1) case labels and the terminal
vertices the labels πv1 (e1 ) = 0 and πvm+1 (em ) = 2. From (4.3) and (4.4), Ω(π) assigns the
terminal vertices the labels Ω(π)v1 (e1 ) = 2 and Ω(π)vm+1 (em ) = 0. We define Ω(π) to assign
the primary case labels to the vertices in Vin (t), which are consistent with the primary edge
weights assigned by WS∆SC0 .
The overall signs of π and Ω(π) for ∪{Vin (t) : t ∈ TC 0 } are the same for π, since the
overall sign for the vertices in Vin (t) is the same for primary case and secondary odd case
labels, and for primary case and secondary even (option 1) case labels, by the proof of
Lemma 4.2.4. The labellings π ∗ and Ω(π)∗ of G∗ differ by a transposition at every vertex of
C by (4.3), (4.4) and (4.5). Finally, for all other vertices v of G the signs of πv∗ and Ω(π)∗v
are the same since these star labellings are the same by (4.2). Thus Ω is sign reversing as
sgn(π ∗ ) = − sgn(Ω(π)∗ ) by Lemma (4.2.4). Finally, Ω(Ω(π)) = π, and so Ω is an involution.
CHAPTER 4. CUBIC PLANAR GRAPHS
45
Since Ω is a sign reversing involution on the set Π1 , it must contain an equal number of
positive and negative star labellings of G.
Proof of Theorem 1.0.4 Consider G with the set of edge weightings W. Then if
Π = Π0 ∪ Π1 is the set of star labelling as defined at the beginning of Section 4.2.6,
X
sgn(π) =
π∈Π
X
π∈Π
X
X
π∈Π0
sgn(π) =
X
sgn(Π) +
X
sgn(Π)
π∈Π1
sgn(Π) + 0
(by Lemma 4.2.6)
π∈Π0
sgn(π) = |Π|
(by Lemma 4.2.5)
π∈Π
X
sgn(π) ≥ 1
(since ρ ∈ Π0 , by Lemma 4.2.1).
π∈Π
So the number of positive star labellings in Π is greater than the number of negative star
labellings in Π. By Corollary 2.0.7, G is (WS 0 + 1)-edge-choosable, for some WS 0 ∈ W. The
edge weighting has WS 0 (e) ≤ 3 for all e ∈ E(G), so (WS 0 + 1) ≤ (4, 4, . . . , 4). Only edges
of G which receive edge weight 3 under WS 0 need to be assigned a colour list of size 4.
For any edge weighting in W, we consider the weights in E(H) ∪ B(H) for
H ∈ Comp(G \ B). For H of Type (1), exactly one edge in E(H) ∪ B(H) has weight 3 and
B(H) contains at least two cut edges. For H of Type (2), exactly one edge in E(H) ∪ B(H)
has weight 3 and B(H) contains at least one cut edge. For H of Type (3), all non-thread
edges have weight 2. For a non-root thread t of H, at most two edges in E(t) ∪ B(t) have
edge weight 3 while B(t) has at least one cut edge. For all three types, each cut edge in
B(H) is also above the root thread t̂ of some other Ĥ ∈ Comp(G \ B), and exactly one
edge in E(t̂) has weight 3. In summary, if G has c cut edges, then WS 0 (e) + 1 = 4 for at
most 3c edges e of G.
Bibliography
[1] N. Alon and M. Tarsi. Colorings and orientations of graphs. Combinatorica, 12(2):125–
134, 1992.
[2] Noga Alon. Restricted colorings of graphs. In Surveys in combinatorics, 1993 (Keele),
volume 187 of London Math. Soc. Lecture Note Ser., pages 1–33. Cambridge Univ.
Press, Cambridge, 1993.
[3] Noga Alon. Combinatorial Nullstellensatz. Combin. Probab. Comput., 8(1-2):7–29,
1999. Recent trends in combinatorics (Mátraháza, 1995).
[4] B. Bollobás and A. J. Harris. List-colourings of graphs. Graphs Combin., 1(2):115–127,
1985.
[5] Arthur A. Drisko. On the number of even and odd Latin squares of order p + 1. Adv.
Math., 128(1):20–35, 1997.
[6] M. N. Ellingham and Luis Goddyn. List edge colourings of some 1-factorable multigraphs. Combinatorica, 16(3):343–352, 1996.
[7] Fred Galvin. The list chromatic index of a bipartite multigraph. J. Combin. Theory
Ser. B, 63(1):153–158, 1995.
[8] Roland Häggkvist and Amanda Chetwynd. Some upper bounds on the total and list
chromatic numbers of multigraphs. J. Graph Theory, 16(5):503–516, 1992.
[9] Roland Häggkvist and Jeannette Janssen. New bounds on the list-chromatic index of
the complete graph and other simple graphs. Combin. Probab. Comput., 6(3):295–313,
1997.
[10] Jeannette C. M. Janssen. The Dinitz problem solved for rectangles. Bull. Amer. Math.
Soc. (N.S.), 29(2):243–249, 1993.
[11] Tommy R. Jensen and Bjarne Toft. Graph Coloring Problems. Wiley-Interscience:
Series in Discrete Mathematics and Optimization. John Wiley & Sons: New York,
Chichester, Brisbane, Toronto, Singapore, 1995.
46
BIBLIOGRAPHY
47
[12] T. Schönberger. Ein beweis des petersenschen graphensatzes. Acta Scientia Mathematica Szeged, 7:51–57, 1934.
[13] P.G. Tait. On the colouring of maps. Proc. Roy. Soc. Edinburgh Sect. A, 10:501–
503,729,1878–80.
[14] V. G. Vizing. Coloring the vertices of a graph in prescribed colors. Diskret. Analiz, (29
Metody Diskret. Anal. v Teorii Kodov i Shem):3–10, 101, 1976.
[15] Douglas B. West. Introduction to Graph Theory. Prentice Hall: Upper Saddle River,
NJ, second edition, 2001.