Tight Bounds on the Chromatic Edge Stability Index of Graphs

TL;DR

This paper establishes optimal upper bounds on the chromatic edge stability index of graphs, providing exact formulas for bipartite graphs and exploring properties of minimum mitigating sets.

Contribution

It introduces tight bounds on the chromatic edge stability index based on vertex degrees and characterizes minimum mitigating sets in relation to vertex degrees.

Findings

Best-possible upper bounds for Class 2 graphs.

Exact expression for bipartite graphs involving maximum matchings.

Not all minimum mitigating sets meet the degree condition.

Abstract

The chromatic edge stability index $\mathrm{es}_{\chi'}(G)$ of a graph $G$ is the minimum number of edges whose removal results in a graph with smaller chromatic index. We give best-possible upper bounds on $\mathrm{es}_{\chi'}(G)$ in terms of the number of vertices of degree $\Delta(G)$ (if $G$ is Class 2), and the numbers of vertices of degree $\Delta(G)$ and ${\Delta(G)-1}$ (if $G$ is Class 1). If $G$ is bipartite we give an exact expression for $\mathrm{es}_{\chi'}(G)$ involving the maximum size of a matching in the subgraph induced by vertices of degree $\Delta(G)$. Finally, we consider whether a minimum mitigating set, that is a set of size $\mathrm{es}_{\chi'}(G)$ whose removal reduces the chromatic index, has the property that every edge meets a vertex of degree at least $\Delta(G)-1$; we prove that this is true for some minimum mitigating set of $G$, but not necessarily for every minimum mitigating set of $G$.