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.