Critical graphs for the chromatic edge-stability number

The chromatic- edge-stability number es(chi) (G) of a graph G is the minimum number of edges whose removal results in a spanning subgraph G' with chi(G') = chi(G) - 1. Edge-stability critical graphs are introduced as the graphs G with the property that es(chi) (G- e) < es(chi) (G) holds for every edge e. E(G). If G is an edge-stability critical graph with.(G) = k and es(chi) (G) = l, then G is (k, l)-critical. Graphs which are (3, 2)-critical and contain at most four odd cycles are classified. It is also proved that the problem of deciding whether a graph G has chi(G) = k and is critical for the chromatic number can be reduced in polynomial time to the problem of deciding whether a graph is (k, 2)-critical. (C) 2020 Elsevier B.V. All rights reserved.