Graphs with girth $2\ell$ and without longer even holes are $3$-colorable

Abstract

For a number $\ell\geq 2$, let $\mathcal{H}_{\ell}$ denote the family of graphs which have girth $2\ell$ and have no even hole with length greater than $2\ell$. Wu, Xu, and Xu conjectured that every graph in $\bigcup_{\ell\geq2}\mathcal{H}_{\ell}$ is 3-colorable. In this paper, we prove that every graph in $\mathcal{H}_{\ell}$ is 3-colorable for any integer $\ell\geq5$.