On critical graphs for the chromatic edge-stability number

Abstract

The chromatic edge-stability number es χ (G) of a graph G is the minimum number of edges whose removal results in a spanning subgraph with the chromatic number smaller than that of G. A graph G is called (3, 2)-critical if χ(G) = 3, es χ (G) = 2 and for any edge e ∈ E(G), es χ (G − e) < es χ (G). In this paper, we characterize (3, 2)critical graphs which contain at least five odd cycles. This answers a question proposed by Brešar, Klavžar and Movarraei in [Critical graphs for the chromatic edge-stability number, Discrete Math. 343(2020) 111845].