  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].
