On the 𝑘-residue of disjoint unions of graphs with
applications to 𝑘-independence
David Amos, Randy Davila, and Dr. Ryan Pepper
Abstract
The 𝑘-residue of a graph, introduced by Jelen in a 1999 paper, is a lower bound on the 𝑘-independence number for every positive integer 𝑘. This generalized earlier work by Favaron, Mahéo and
Saclé, by Griggs and Kleitman, and also by Triesch, who all showed that the independence number of a graph is at least as large as its Havel-Hakimi residue, defined by Fajtlowicz. We show here
that, for every positive integer 𝑘, the 𝑘-residue of disjoint unions is at most the sum of the 𝑘-residues of the connected components considered separately, and give applications of this lemma. Our
main application is an improvement on Jelen’s bound for connected graphs which have a maximum degree cut-vertex.
Preliminaries
Main Results
Applications of the
Disjoint Union Lemma
Let 𝐺 = 𝑉, 𝐸 be a simple, finite and undirected graph of order 𝑛 with vertex set 𝑉
and edge set 𝐸. The independence number of 𝐺, denoted 𝛼 𝐺 , is the cardinality of a
largest set of pairwise non-adjacent vertices. The degree sequence of 𝐺, denoted 𝐷(𝐺),
is the sequence of degrees of each vertex of 𝑉, where the degree of a vertex is the
number of vertices that it is adjacent to. We use Δ to denote the largest degree, and
assume that 𝐷 is ordered in non-increasing order.
Lemma (Pepper [8]). Let 𝐺 be a graph 𝑎𝑛𝑑 𝑘 ∈ ℤ+ . If 𝑘 ≥ Δ,
then
Theorem ([1]). Let 𝐺 be a disconnected graph with components 𝐺1 , 𝐺2 , … , 𝐺𝑝 and let
𝑘 ∈ ℤ+ . Then
𝑚
𝑅𝑘 𝐺 = 𝑛 − ,
𝑘
The elimination sequence of 𝐷 is the sequence of terms deleted, in order, during the
HHP, together with the zeros obtained at the end of the HHP.
𝑅𝑘 𝐺 ≤
𝑝
𝑅𝑘 𝐺𝑖 ≤
𝑖=1
where 𝑚 is the number of edges in 𝐺.
Lemma ([1]). Let 𝐾𝑛 be the complete graph on n vertices and
let 𝑘 ∈ ℤ+ . Then
𝑅𝑘 𝐾𝑛 =
𝑛
𝑘+1
,
2
𝑛 𝑛−1
− 2𝑘
𝛼𝑘 𝐺𝑖 = 𝛼𝑘 𝐺 .
𝑖=1
p
′
The Havel-Hakimi derivative of 𝐷 is the sequence 𝐷 obtained by removing a largest
term Δ from D and reducing by one the next Δ highest terms. A well known theorem of
Havel and Hakimi states that 𝐷 is graphic if and only if 𝐷′ is graphic [5, 6]. As a
consequence of this, repeatedly taking Havel-Hakimi derivatives of the degree
sequence of a graph will eventually terminate in a sequence of zeros. This process is
called the Havel-Hakimi Process (HHP) and the number of remaining zeros is called
the residue. The residue was conjectured to be a lower bound for the independence
number in [2] and this conjecture was later proven in [3,4,7,9]
𝑝
𝑖𝑓 𝑘 ≤ 𝑛
The difference between 𝑅𝑘 (𝐺) and i=1 R k Gi can be arbitrarily large (see Figure 1
for a small example where the difference is 3 for 𝑅1 ).
Theorem ([1]). Let 𝐺 be a graph with a maximum degree cut vertex 𝑐 and let
𝐺1 , 𝐺2 , … , 𝐺𝑝 be the components of 𝐺 − 𝑐 . Then, for 𝑘 ∈ ℤ+ , 𝑘 ≤ Δ,
.
, 𝑖𝑓𝑘 ≥ 𝑛 + 1
Theorem (The Disjoint Union Lemma [1]). Let 𝐺 and 𝐻 be
any two graphs and let 𝐺 ∪ 𝐻 be their disjoint union. Then
𝑅𝑘 𝐺 ∪ 𝐻 ≤ 𝑅𝑘 𝐺 + 𝑅𝑘 𝐻 .
𝑝
𝑅𝑘 𝐺 ≤
𝑝
𝑅𝑘 𝐺𝑖 ≤
𝑖=1
𝛼𝑘 𝐺𝑖 ≤ 𝛼𝑘 𝐺 .
𝑖=1
𝑝
Again, the difference between 𝑅𝑘 (𝐺) and 𝑖=1 𝑅𝑘 𝐺𝑖 can be arbitrarily large (see
Figure 2 for a small example where the difference is 5 for 𝑅1 ).
Let 𝑘 be a positive integer. The 𝑘-residue of a graph 𝐺, first introduced by Jelen [7], is
defined as
1
𝑅𝑘 𝐺 =
𝑘
𝑘−1
𝑘 − 𝑖 𝑓𝑖
𝑖=0
where 𝑓𝑖 is the frequency with which 𝑖 occurs in the elimination sequence of 𝐷. Note
that 𝑅1 is equivalent to the residue. The 𝑘-residue is a lower bound for the 𝑘independence number 𝛼𝑘 of a graph [7] – that is, the cardinality of a largest subset of
vertices that induces a subgraph of maximum degree at most 𝑘 − 1 (observe that
𝛼1 = 𝛼).
Figure 2
Open Problems
Problem 1: Characterize the case of equality for the Disjoint Union Lemma.
Problem 2: Find more applications of the Disjoint Union Lemma.
Problem 3: Find a formula for the 𝑘-Residue of regular graphs.
Theorem (Jelen [6]). Let 𝐺 be a graph. Then for every positive integer 𝑘,
𝑅𝑘 𝐺 ≤ 𝛼𝑘 𝐺 .
Figure 1
Department of Computer and Mathematical Sciences
www.uhd.edu/cms
References
[1] D. Amos, R. Davila, R. Pepper. On the 𝑘-residue of disjoint unions of graph with
applications to 𝑘-independence (submitted for publication).
[2] S. Fajtlowicz, On the conjectures of Graffiti, III, Congressus Numerantum 66
(1988), 23-32.
[3] O. Favaron, M. Mahéo, and J.F. Saclé, On the residue of a graph, J. Graph Theory 15
(1991), 39-64.
[4] J. R. Griggs and D. J. Kleitman, Independence and the Havel-Hakimi residue,
Discrete Math 127 (1994), 241-249.
[5] 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.
[6] V. Havel, A remark on the existence of finite graphs, Casopis Pest Mat 80 (1955),
477-480 (Czech).
[7] F. Jelen, k-independence and the k-residue of a graph, J. Graph Theory 127 (1999),
209-212.
[8] R. Pepper, Binding independence, Ph.D. thesis, University of Houston, 2004.
[9] E. Triesch, Degree sequences of graphs and dominance order, J. Graph Theory 22
(1996), 89-93.