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arXiv · 2206.03953

Tight Bounds on the Chromatic Edge Stability Index of Graphs

Saieed Akbari·John Haslegrave·Mehrbod Javadi·Nasim Nahvi·Helia Niaparast
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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 $G$.

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Saieed Akbari, John Haslegrave, Mehrbod Javadi, Nasim Nahvi, Helia Niaparast. 2022-06-08. Tight Bounds on the Chromatic Edge Stability Index of Graphs. https://doi.org/10.1016/j.disc.2023.113850

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