Residue and the independence number of a graph
David Amos
Texas A&M University
amosd2@math.tamu.edu
April 3, 2014
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Graphic sequences
Definition (Graphic sequence)
The degree sequence of a graph G , denoted D(G ), is the sequence of
degrees of each vertex of G in non-increasing order. A non-increasing
finite sequence d of non-negative integers is graphic if d = D(G ) for some
graph G .
Example
Notation: {3, 13 } represents the sequence 3, 1, 1, 1.
The sequence {3, 13 } is graphic.
The sequence {2, 1} is not graphic.
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Characterizations of graphic sequences
Theorem (Erdös and Gallai, 1960)
A non-increasing
Pn sequence {d1 , d2 , . . . , dn } of positive integers is graphic if
and only if i=1 di is even and the sequence obeys the property
r
X
di ≤ r (r − 1) +
i=1
n
X
min(r , di )
i=r +1
for each positive integer r ≤ n − 1.
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Characterizations of graphic sequences
Theorem (Havel, 1955; Hakimi, 1962)
A non-increasing sequence D = {d1 , d2 , . . . , dn } of non-negative integers is
graphic if and only if d1 ≤ n − 1 and the sequence
D 0 = {d2 − 1, d3 − 1, . . . , dd1 +1 − 1, dd1 +2 , . . . , dn }
is graphic. The derived sequence D 0 is called the Havel-Hakimi
derivative of D.
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The Havel-Hakimi Process
Let D = {d1 , d2 , . . . , dn } be a non-increasing sequence of non-negative
integers with d1 ≤ n − 1 and D 1 its Havel-Hakimi derivative.
Let D 2 be the Havel-Hakimi derivative of D 1 , D 3 the Havel-Hakimi
derivative of D 2 , and so on.
In general D i is the Havel-Hakimi derivative of D i−1 , 1 ≤ i ≤ m, where we
set D 0 = D and m ≤ n − 1 is the first index for which either D m is the
empty sequence, or the largest element of D m is non-positive.
It can be shown that D is graphic if and only if D m consists only of zeros.
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Residue
Definition (Fajtlowicz, 1988)
Let D be a graphic sequence. The number of zeros in the final sequence
obtained by the Havel-Hakimi process beginning with D is called the
residue of D and is denoted R(D). Sometimes we abuse notation and
write R(G ) for the residue of the degree sequence of G .
Example
R({3, 13 }) = 3.
R({45 }) = 1.
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Independence number
Definition
An independent set in a graph G is a subset S of vertices of G with the
property that no two vertices in S lie on the same edge in G .
The independence number of G , denoted α(G ), is the cardinality of a
largest independent set in G .
Example
The independence number of a complete graph is 1 (the converse is also
true).
The independence number of any n-vertex cycle is b n2 c.
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Importance of the independence number
Independent sets play an important role in Ramsey theory for graphs.
There is evidence that the independence number of the graph of certain
classes of hydrocarbons is a measure of chemical stability [Fajtlowicz and
Larson, 2003].
The independent set decision problem is NP-complete [Garey and Johnson,
1979], and a large body of research has focused on lower and upper
bounds for the independence number of a graph.
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MAXINE
MAXINE is a greedy algorithm, introduced by Johnson (1974) and Griggs
(1983), and given the name MAXINE by Fajtlowicz, for producing an
independent set in a graph.
Let G be a graph and v a maximum degree vertex of G . Remove v and
call the new graph H. Then remove a maximum degree vertex from H.
Continue this process until the resulting graph has no edges.
Theorem
Let G be a graph and M(G ) the largest number of vertices in the final
graph obtained by any performance of MAXINE on G . Then
M(G ) ≤ α(G ).
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Residue and independence number
Theorem
For any graph G , R(G ) ≤ α(G ).
Conjectured by Fajtlowicz, via his conjecture-making computer
program GRAFFITI [Fajtlowicz, 1988].
First proven by Favaron, Mahéo and Saclé, with a rather complicated
and technical proof [Favaron et al., 1991].
A second, more natural proof appeared in 1992 [Griggs and Kleitman,
1992]. They showed that R(G ) ≤ M(G ).
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The k-independence number
Definition (Fink and Jacobson, 1985; Hopkins and Staton, 1986)
Let G be a graph and k a positive integer. A subset S of vertices of G is
called a k-independent set if the subgraph of G induced by S has
maximum degree strictly less than k.
The k-independence number of G , denoted αk (G ), is the cardinality of
a largest k-independent set of G . Note that α1 (G ) = α(G ).
Example
The k-independence number of an n-vertex complete graph is min{k, n}.
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The elimination sequence
Definition (Triesch, 1996)
Let D be a graphic sequence. The elimination sequence of D is the
sequence of integers deleted at each step of the Havel-Hakimi process
beginning with D, followed by the zeros obtained in the final step of the
Havel-Hakimi process.
Example
The elimination sequence of {3, 13 } is {3, 03 }.
The elimination sequence of {32 , 24 } is {3, 2, 12 , 02 }.
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Jelen’s k-residue
Definition (Jelen, 1999)
Let D = {d1 , d2 , . . . , dn } be a graphic sequence and E the elimination
sequence of D. For any positive integer k, the k-residue of G , denoted
Rk (D), is given by
k−1
1X
Rk (D) =
(k − i)fi ,
k
i=0
where fi is the frequency of i in E . Note that R1 (D) = R(D).
Jelen’s motivation for the definition of k-residue was to prove the
following:
Theorem (Jelen, 1999)
For any graph G , Rk (G ) ≤ αk (G ).
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Properties of k-residue
Theorem (Pepper, 2004)
Let G = (V , E ) be a graph, ∆ the maximum degree of G , and k a
positive integer. If k ≥ ∆, then
Rk (G ) = n −
m
,
k
where n = |V | and m = |E |.
Theorem (ADP, 2014)
Let G = (V , E ) be a graph. If v ∈ V is any maximum degree vertex of G ,
then
Rk (G ) ≤ Rk (G − v ),
where G − v is the graph obtained from G by removing v .
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The Disjoint Union Lemma
Theorem (The Disjoint Union Lemma; ADP, 2014)
Let G and H be any two graphs and G ∪ H their disjoint union. Then
Rk (G ∪ H) ≤ Rk (G ) + Rk (H).
The above theorem can be extended to the disjoint union of any finite
number of graphs by an easy induction argument.
Theorem (ADP, 2014)
Let G1 , G2 , . . . , Gp , p ≥ 2, be any graphs and k a positive integer. Then
Rk
p
[
i=1
David Amos (TAMU)
p
X
Gi ≤
Rk (Gi ).
i=1
Residue and independence number
April 3, 2014
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Applications of the Disjoint Union Lemma
Theorem (ADP, 2014)
Let G be a graph with connected components G1 , G2 , . . . , Gp and k a
positive integer. Then
Rk (G ) ≤
p
X
Rk (Gi ) ≤ αk (G ).
i=1
Theorem (ADP, 2014)
Let G be a graph with a maximum degree cut vertex v , G1 , G2 , . . . , Gp the
connected components of G − v , ∆ the maximum degree of G , and k a
positive integer. If k ≤ ∆, then
Rk (G ) ≤
p
X
Rk (Gi ) ≤ αk (G ).
i=1
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Open problems
Definition
A graph G is said to be r -regular if every vertex of G has degree r .
Theorem (Favaron et al., 1991)
n
If G = (V , E ) is r -regular, then R(G ) = d r +1
e.
Problem
Is there a simple formula for the k-residue of an n-vertex r -regular graph in
terms of n and r , where n is the number of vertices of G ?
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Open problems
Problem
Characterize equality for the Disjoint Union Lemma.
Note that this problem really is about graphs, and not just degree
sequences. For example, consider the graphic sequence {28 }, which has
two different disconnected realizations: 1) as two copies of a cycle on 4
vertices, and 2) as the disjoint union of a cycle on 5 vertice and a cycle on
3 vertices. In the first case, equality is not achieved when k = 1, but it is
in the second.
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Open problems
Problem
Characterize the graphs G for which αk (G ) = Rk (G ).
The problem is open even when k = 1. In fact, no characterization is
known even for α(G ) = R(G ) = 2. It is known, however, that
α(G ) = R(G ) = 1 if and only if G is a complete graph.
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References
D. Amos, R. Davila and R. Pepper, On the k-residue of disjoint unions of graphs with
applications on k-independence, Discrete Mathematics, 321 (2014), 24-34.
S. Fajtlowicz, On the conjectures of Graffiti, III, Congressus Numerantium 66 (1988),
23–32.
S. Fajtlowicz and C. Larson, Graph-Theoretic Independence as a Predictor of Fullerene
Stability, Chemical Physics Letters, 377 (2003), 485-490.
O. Favaron, M. Mahéo, and J.F. Saclé, On the residue of a graph, Journal of Graph
Theory 15 (1991), 39–64.
M. Garey and D. Johnson, Computers and intractability, W. H. Freeman and Company,
New York, 1979.
J. R. Griggs and D. J. Kleitman, Independence and the Havel-Hakimi residue, Discrete
Mathematics 127 (1999), 241–249.
J.R. Griggs, Lower bounds on the independence number in terms of degrees, Journal of
Combinatorial Theory, Series B 34 (1983), 22–39.
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References
S. L. Hakimi, On the realizability of a set of integers as degrees of the vertices of a
linear graph, I., SIAM J. Appl. Math 10 (1962), 496-506.
V. Havel, A remark on the existence of finite graphs, Casopis Pest Mat 80 (1955),
477–480 (Czech).
G. Hopkins and W. Staton, Vertex partitions and k-small subsets of graphs, Ars
Combinatorics 22 (1986), 19–24.
F. Jelen, k-independence and the k-residue of a graph, Journal of Graph Theory 127
(1999), 209–212.
D.S. Johnson, Approximation algorithms for combinatorial problems, J. Comput. System
Sci. 9 (1974), 256-278.
R. Pepper, Binding independence, Ph.D. Thesis, University of Houston, 2004.
E. Triesch, Degree sequences of graphs and dominance order, Journal of Graph Theory
22 (1996), 89-93.
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