On the chromatic vertex stability number of graphs
Introduction
Throughout this paper all graphs are finite, simple, and having at least one edge. The chromatic edge stability number of a graph is the minimum number of edges whose deletion results in a graph with . This natural coloring concept was introduced in 1980 by Staton [12], and independently rediscovered much later in [5]. Nevertheless, this concept has become the subject of wider interest only recently. The paper [10] gives, among other results, a characterization of graphs with equal chromatic edge stability number and chromatic bondage number. In [7], edge-stability critical graphs were studied. The paper [2] brings Nordhaus–Gaddum type inequality for (stronger than a related result from [5]), sharp upper bounds on in terms of size and of maximum degree, and a characterization of graphs with among -regular graphs for . In [8] progress on three open problems from [2] are reported. The chromatic edge stability number has been generalized to arbitrary graphical invariants in [9], where in particular it was considered with respect to the chromatic index, see also [1], [3].
Like edge stability numbers, vertex stability numbers were introduced in the 1980s or earlier. In [6], the -stability of a graph , where is an arbitrary graph invariant, is defined as the minimum number of vertices whose removal changes . The paper [6] then proceeds by investigating the stability with respect to the domination number and the independence number, which in turn led to a series papers investigation the stability with respect to these two invariants. In this paper, however, we are interested in the stability with respect to the chromatic number. At least as far as we know, this concept has not yet been explored (in [6], the stability with respect to the chromatic number is briefly mentioned only in one sentence) which we find quite surprising since vertex versions are usually considered before edge versions. The closest investigation we are aware of is the paper [4], where the stability with respect to the distinguishing number is investigated.
Let be a graph. The chromatic vertex stability number of is the minimum number of vertices of such that their deletion results in a graph with . For instance, it is straightforward to see that , where is the Petersen graph. Note that if , then is just the minimum cardinality of a set such that the graph induced by is bipartite.
We also introduce the independent chromatic vertex stability number, , of as the minimum number of independent vertices such that their deletion results in a graph with . Then our main result reads as follows.
Theorem 1.1 If is a graph with , then .
The paper is structured as follows. In the rest of this section we recall needed definitions and concepts. In Section 2 we prove Theorem 1.1. In the subsequent section we show that Theorem 1.1 need not hold for graphs with , and discuss a possible threshold function that would guarantee that if , then . In the final section we prove that if , then , and give a Nordhaus–Gaddum-type result on the chromatic vertex stability number.
Given a graph , a function with for each edge is a proper -coloring of . The minimum for which admits a proper -coloring is the chromatic number of . If is a proper coloring of , then the set of all vertices of with color , , is a color class and will be denoted by . The open neighborhood of a vertex in is the set of neighbors of , denoted by , whereas the closed neighborhood of is . The degree of a vertex in is denoted by . The subgraph of induced by will be denoted by . The complete graph of order is denoted by and the complement of a graph by . Finally, the order of will be denoted by .
Section snippets
Proof of Theorem 1.1
The following lemma follows directly from Brooks’ Theorem (cf. [13, p. 197]).
Lemma 2.1 Let be a connected graph with . If , then , and if , then for some odd .
For the proof of the theorem, we also need the following lemma.
Lemma 2.2 Let be a connected graph with and . Then there exists such that and .
Proof Let . Since , we have . If , then there exists such that , as desired.
Discussion on graphs with smaller chromatic number
In the next remark we demonstrate that Theorem 1.1 does not extend to the case when .
Remark 3.1 Let , , , be the graph obtained from the cycle as follows. For each vertex of the cycle take two disjoint complete graphs , select a fixed vertex in each of them, and identify the two vertices with . Then it is straightforward to verify that , , , and .
In view of Remark 3.1 we pose the next problem for which we feel the answer is yes.
Problem 3.2 Is it
Relation with and a Nordhaus–Gaddum-type result
In this final section we first prove that if the chromatic number of a graph is large, then its chromatic vertex stability number is equal to its chromatic edge stability number. More precisely, we have the following result.
Theorem 4.1 If is a graph with , then .
Proof Clearly, , hence we need to show that . We proceed by induction on . The base case is when . Let be a vertex such that . Let be a proper -coloring of .
Acknowledgments
We thank one of the reviewers for careful reading of the article and for many very helpful comments.
The research of the first author was supported by Grant Number G981202 from Sharif University of Technology. Sandi Klavžar acknowledges the financial support from the Slovenian Research Agency (research core funding P1-0297 and projects N1-0095, J1-1693, J1-2452).
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