Some extremal results on the chromatic-stability index

TL;DR

This paper investigates the extremal properties of the chromatic-stability index in graphs, addressing open problems, providing counterexamples, characterizations, and conditions related to the index and graph colorings.

Contribution

It constructs counterexamples to known characterizations, characterizes graphs with specific stability sum properties, and establishes sufficient conditions for bounds on the index.

Findings

Counterexamples for k-regular graphs with es_χ(G)=1 for k≥6.

Characterization of graphs with es_χ(G)+es_χ(Ḡ)=2 when χ(G)=3.

Necessary and sufficient conditions for graphs attaining upper bounds on es_χ(G).

Abstract

The $\chi$-stability index ${\rm es}_{\chi}(G)$ of a graph $G$ is the minimum number of its edges whose removal results in a graph with the chromatic number smaller than that of $G$. In this paper three open problems from [European J.\ Combin.\ 84 (2020) 103042] are considered. Examples are constructed which demonstrate that a known characterization of $k$-regular ($k\le 5$) graphs $G$ with ${\rm es}_{\chi}(G) = 1$ does not extend to $k\ge 6$. Graphs $G$ with $\chi(G)=3$ for which ${\rm es}_{\chi}(G)+{\rm es}_{\chi}(\overline{G}) = 2$ holds are characterized. Necessary conditions on graphs $G$ which attain a known upper bound on ${\rm es}_{\chi}(G)$ in terms of the order and the chromatic number of $G$ are derived. The conditions are proved to be sufficient when $n\equiv 2 \pmod 3$ and $\chi(G)=3$.