Isolation of Regular Graphs and k-Chromatic Graphs

Abstract

Given a set \({\mathcal {F}}\) of graphs, we call a copy of a graph in \({\mathcal {F}}\) an \({\mathcal {F}}\)-graph. The \({\mathcal {F}}\)-isolation number of a graph G, denoted by \(\iota (G,{\mathcal {F}})\), is the size of a smallest set D of vertices of G such that the closed neighborhood of D intersects the vertex sets of the \({\mathcal {F}}\)-graphs contained by G (equivalently, \(G - N[D]\) contains no \({\mathcal {F}}\)-graph). Thus, \(\iota (G,\{K_1\})\) is the domination number of G. For any integer \(k \ge 1\), let \({\mathcal {F}}_{1,k}\) be the set of regular graphs of degree at least \(k-1\), let \({\mathcal {F}}_{2,k}\) be the set of graphs whose chromatic number is at least k, and let \({\mathcal {F}}_{3,k}\) be the union of \({\mathcal {F}}_{1,k}\) and \({\mathcal {F}}_{2,k}\). Thus, k-cliques are members of both \({\mathcal {F}}_{1,k}\) and \({\mathcal {F}}_{2,k}\). We prove that for each \(i \in \{1, 2, 3\}\), \(\frac{m+1}{{k \atopwithdelims ()2} + 2}\) is a best possible upper bound on \(\iota (G, {\mathcal {F}}_{i,k})\) for connected m-edge graphs G that are not k-cliques. The bound is attained by infinitely many (non-isomorphic) graphs. The proof of the bound depends on determining the graphs attaining the bound. This appears to be a new feature in the literature on isolation. Among the result’s consequences are a sharp bound of Fenech, Kaemawichanurat, and the present author on the k-clique isolation number and a sharp bound on the cycle isolation number.