Isolation of Regular Graphs and k-Chromatic Graphs

Abstract

AbstractGiven a set $${\mathcal {F}}$$ F of graphs, we call a copy of a graph in $${\mathcal {F}}$$ F an $${\mathcal {F}}$$ F -graph. The $${\mathcal {F}}$$ F -isolation number of a graph G, denoted by $$\iota (G,{\mathcal {F}})$$ ι ( G , 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 $${\mathcal {F}}$$ F -graphs contained by G (equivalently, $$G - N[D]$$ G - N [ D ] contains no $${\mathcal {F}}$$ F -graph). Thus, $$\iota (G,\{K_1\})$$ ι ( G , { K 1 } ) is the domination number of G. For any integer $$k \ge 1$$ k ≥ 1 , let $${\mathcal {F}}_{1,k}$$ F 1 , k be the set of regular graphs of degree at least $$k-1$$ k - 1 , let $${\mathcal {F}}_{2,k}$$ F 2 , k be the set of graphs whose chromatic number is at least k, and let $${\mathcal {F}}_{3,k}$$ F 3 , k be the union of $${\mathcal {F}}_{1,k}$$ F 1 , k and $${\mathcal {F}}_{2,k}$$ F 2 , k . Thus, k-cliques are members of both $${\mathcal {F}}_{1,k}$$ F 1 , k and $${\mathcal {F}}_{2,k}$$ F 2 , k . We prove that for each $$i \in \{1, 2, 3\}$$ i ∈ { 1 , 2 , 3 } , $$\frac{m+1}{{k \atopwithdelims ()2} + 2}$$ m + 1 k 2 + 2 is a best possible upper bound on $$\iota (G, {\mathcal {F}}_{i,k})$$ ι ( G , 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.