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

TL;DR

This paper proves that certain $k$-critical graphs with a limited number of degree-2 vertices always contain an even factor, advancing understanding of their structural properties.

Contribution

It establishes that $k$-critical graphs with up to $2k-6$ degree-2 vertices necessarily have an even factor, addressing a question about their structural characteristics.

Findings

$k$-critical graphs with ≤ $2k-6$ degree-2 vertices have an even factor.

Provides a partial answer to a question posed by Bej and the first author.

Enhances understanding of the structure of edge-chromatic-critical graphs.

Abstract

A finite simple connected graph $G$ with maximum degree $k$ is $k$-critical if 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 graph with at most $2k-6$ vertices of degree 2 has an even factor.