Tight Bounds on the Chromatic Edge Stability Index of Graphs

Akbari, Saieed and Haslegrave, John and Javadi, Mehrbod and Nahvi, Nasim and Niaparast, Helia (2024) Tight Bounds on the Chromatic Edge Stability Index of Graphs. Discrete Mathematics, 347 (4): 113850. ISSN 0012-365X

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Abstract

The chromatic edge stability index es χ ′ (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 es χ ′ (G) in terms of the number of vertices of degree Δ(G) (if G is Class 2), and the numbers of vertices of degree Δ(G) and Δ(G)−1 (if G is Class 1). If G is bipartite we give an exact expression for es χ ′ (G) involving the maximum size of a matching in the subgraph induced by the vertices of degree Δ(G). Finally, we consider whether a minimum mitigating set, that is a set of size es χ ′ (G) whose removal reduces the chromatic index, has the property that every edge meets a vertex of degree at least Δ(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.