On critical graphs for the chromatic edge-stability number

The {\em chromatic edge-stability number} $es_{\chi}(G)$ 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 {\em $(3,2)$-critical} if $\chi(G)=3$, $es_{\chi}(G)=2$ and for any edge $e\in E(G)$, $es_{\chi}(G-e)<es_{\chi}(G)$. In this paper, we characterize $(3,2)$-critical graphs which contain at least five odd cycles. This answers a question proposed by Bre\v{s}ar, Klav\v{z}ar and Movarraei in [Critical graphs for the chromatic edge-stability number, {\it Discrete Math.} {\bf 343}(2020) 111845].