On the Chromatic Vertex Stability Number of Graphs

TL;DR

This paper investigates the chromatic vertex stability number of graphs, establishing conditions under which it equals the independent chromatic vertex stability number or the chromatic edge stability number, and provides related bounds and a Nordhaus-Gaddum-type result.

Contribution

It proves new equalities between different stability numbers under specific chromatic and maximum degree conditions, advancing understanding of graph stability properties.

Findings

Equality of stability numbers when hi(G) q elta(G) or hi(G)   +1

Counterexamples for graphs with hi(G)  ( +1)/2

A Nordhaus-Gaddum-type inequality for the chromatic vertex stability number

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.