On the chromatic vertex stability number of graphs

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Abstract

The chromatic vertex (resp. edge) stability number vsχ(G) (resp. esχ(G)) of a graph G is the minimum number of vertices (resp. edges) whose deletion results in a graph H with χ(H)=χ(G)1. In the main result it is proved that if G is a graph with χ(G){Δ(G),Δ(G)+1}, then vsχ(G)=ivsχ(G), where ivsχ(G) is the independent chromatic vertex stability number. The result need not hold for graphs G with χ(G)Δ(G)+12. It is proved that if χ(G)>Δ(G)2+1, then vsχ(G)=esχ(G). A Nordhaus–Gaddum-type result on the chromatic vertex stability number is also given.

Introduction

Throughout this paper all graphs are finite, simple, and having at least one edge. The chromatic edge stability number esχ(G) of a graph G is the minimum number of edges whose deletion results in a graph H with χ(H)=χ(G)1. 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 esχ(G) (stronger than a related result from [5]), sharp upper bounds on esχ(G) in terms of size and of maximum degree, and a characterization of graphs with esχ(G)=1 among k-regular graphs for k5. 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 μ-stability of a graph G, where μ is an arbitrary graph invariant, is defined as the minimum number of vertices whose removal changes μ. 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 G be a graph. The chromatic vertex stability number vsχ(G) of G is the minimum number of vertices of G such that their deletion results in a graph H with χ(H)=χ(G)1. For instance, it is straightforward to see that vsχ(P)=3, where P is the Petersen graph. Note that if χ(G)=3, then vsχ(G) is just the minimum cardinality of a set XV(G) such that the graph induced by V(G)X is bipartite.

We also introduce the independent chromatic vertex stability number, ivsχ(G), of G as the minimum number of independent vertices such that their deletion results in a graph H with χ(H)=χ(G)1. Then our main result reads as follows.

Theorem 1.1

If G is a graph with χ(G){Δ(G),Δ(G)+1}, then vsχ(G)=ivsχ(G).

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 G with χ(G)Δ(G)+12, and discuss a possible threshold function f(Δ(G)) that would guarantee that if χ(G)f(Δ(G)), then vsχ(G)=ivsχ(G). In the final section we prove that if χ(G)>Δ(G)2+1, then vsχ(G)=esχ(G), and give a Nordhaus–Gaddum-type result on the chromatic vertex stability number.

Given a graph G=(V(G),E(G)), a function c:V(G)[k]={1,,k} with c(v)c(u) for each edge uv is a proper k-coloring of G. The minimum k for which G admits a proper k-coloring is the chromatic number χ(G) of G. If c is a proper coloring of G, then the set of all vertices of G with color i, i[χ(G)], is a color class and will be denoted by Ci. The open neighborhood of a vertex v in G is the set of neighbors of v, denoted by NG(v), whereas the closed neighborhood of v is NG[v]=NG(v){v}. The degree of a vertex v in G is denoted by dG(v). The subgraph of G induced by AV(G) will be denoted by GA. The complete graph of order n is denoted by Kn and the complement of a graph G by G¯. Finally, the order of G will be denoted by n(G).

Section snippets

Proof of Theorem 1.1

The following lemma follows directly from Brooks’ Theorem (cf. [13, p. 197]).

Lemma 2.1

Let G be a connected graph with χ(G)=Δ(G)+1. If Δ(G)2, then GKΔ(G)+1, and if Δ(G)=2, then GCn for some odd n.

For the proof of the theorem, we also need the following lemma.

Lemma 2.2

Let G be a connected graph with χ(G)=Δ(G) and vsχ(G)=1. Then there exists vV(G) such that dG(v)=Δ(G) and χ(Gv)=Δ(G)1.

Proof

Let S={uV(G):χ(Gu)=Δ(G)1}. Since vsχ(G)=1, we have S. If S=V(G), then there exists uS such that dG(u)=Δ(G), 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 χ(G)Δ(G)2.

Remark 3.1

Let Gn,k, n2, k3, be the graph obtained from the cycle C2n as follows. For each vertex u of the cycle take two disjoint complete graphs Kk, select a fixed vertex in each of them, and identify the two vertices with u. Then it is straightforward to verify that χ(Gn,k)=k, Δ(Gn,k)=2k, vsχ(Gn,k)=2n, and ivsχ(Gn,k)=3n.

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 esχ 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 G is a graph with χ(G)>Δ(G)2+1, then vsχ(G)=esχ(G).

Proof

Clearly, vsχ(G)esχ(G), hence we need to show that vsχ(G)esχ(G). We proceed by induction on vsχ(G).

The base case is when vsχ(G)=1. Let u be a vertex such that χ(Gu)=χ(G)1. Let c be a proper (χ(G)1)-coloring of Gu.

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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