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A finite simple connected graph $G$ with maximum degree $k$ is $k$-criticalif it has chromatic index $\chi'(G)=k+1$ and $\chi'(G-e)=k$ for every edge$e\in E(G)$. Bej and the first author raised the question whether every$k$-critical graph has an even factor. We prove that every $k$-critical graphwith at most $2k-6$ vertices of degree 2 has an even factor.

Even factors in edge-chromatic-critical graphs with a small number of divalent vertices