On the chromatic edge stability index of graphs

Abstract

Given a non-trivial graph $G$, the minimum cardinality of a set of edges $F$ in $G$ such that $χ'(G \setminus F)<χ'(G)$ is called the chromatic edge stability index of $G$, denoted by $es_{χ'}(G)$, and such a (smallest) set $F$ is called a (minimum) mitigating set. While $1\le es_{χ'}(G)\le \lfloor n/2\rfloor$ holds for any graph $G$, we investigate the graphs with extremal and near-extremal values of $es_{χ'}(G)$. The graphs $G$ with $es_{χ'}(G)=\lfloor n/2\rfloor$ are classified, and the graphs $G$ with $es_{χ'}(G)=\lfloor n/2\rfloor-1$ and $χ'(G)=Δ(G)+1$ are characterized. We establish that the odd cycles and $K_2$ are exactly the regular connected graphs with the chromatic edge stability index $1$; on the other hand, we prove that it is NP-hard to verify whether a graph $G$ has $es_{χ'}(G)=1$. We also prove that every minimum mitigating set of an $r$-regular graph $G$, where $r\ne 4$, with $es_{χ'}(G)=2$ is a matching. Furthermore, we propose a conjecture that for every graph $G$ there exists a minimum mitigating set, which is a matching, and prove that the conjecture holds for graphs $G$ with $es_{χ'}(G)\in\{1,2,\lfloor n/2\rfloor-1,\lfloor n/2\rfloor\}$, and for bipartite graphs.