On the Chromatic Vertex Stability Number of Graphs

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.