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.

中文翻译：

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彩色边缘稳定性数的临界图

图 $G$ 的彩色边稳定性数 ${\rm es}_{\chi}(G)$ 是其去除导致生成具有 $\chi(G) 的跨越子图 $G'$ 的最小边数')=\chi(G)-1$。边稳定性临界图被引入为具有 ${\rm es}_{\chi}(Ge) < {\rm es}_{\chi}(G)$ 对每条边成立的属性的图 $G$ $e\in E(G)$。如果 $G$ 是具有 $\chi(G)=k$ 和 ${\rm es}_{\chi}(G)=\ell$ 的边稳定性临界图，则 $G$ 是 $(k, \ell)$-关键。对$(3,2)$-critical 且最多包含四个奇数周期的图进行分类。还证明了判断图$G$是否具有$\chi(G)=k$并且对色数很关键的问题可以在多项式时间内简化为判断图是否为$(k)的问题,2)$-关键。