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arXiv · 2509.01137

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

Rong Chen
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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 $\ell\geq5$.

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Rong Chen. 2025-09-01. Graphs with girth $2\ell$ and without longer even holes are $3$-colorable. https://arxiv.org/abs/2509.01137

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