  The chromatic edge-stability number ${\rm es}_{\chi}(G)$ 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 ${\rm es}_{\chi}(G-e) < {\rm es}_{\chi}(G)$
holds for every edge $e\in E(G)$. If $G$ is an edge-stability critical graph
with $\chi(G)=k$ and ${\rm es}_{\chi}(G)=\ell$, then $G$ is
$(k,\ell)$-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.
