On the Chromatic Vertex Stability Number of Graphs

Abstract

The chromatic vertex (resp.\ edge) stability number ${\rm vs}_{\chi}(G)$ (resp.\ ${\rm es}_{\chi}(G)$) of a graph $G$ is the minimum number of vertices (resp.\ edges) whose deletion results in a graph $H$ with $\chi(H)=\chi(G)-1$. In the main result it is proved that if $G$ is a graph with $\chi(G) \in \{ \Delta(G), \Delta(G)+1 \}$, then ${\rm vs}_{\chi}(G) = {\rm ivs}_{\chi}(G)$, where ${\rm ivs}_{\chi}(G)$ is the independent chromatic vertex stability number. The result need not hold for graphs $G$ with $\chi(G) \le \frac{\Delta(G)+1}{2}$. It is proved that if $\chi(G) > \frac{\Delta(G)}{2}+1$, then ${\rm vs}_{\chi}(G) = {\rm es}_{\chi}(G)$. A Nordhaus-Gaddum-type result on the chromatic vertex stability number is also given.