Graphs with girth $2\ell+1$ and without longer odd holes are $3$-colorable

TL;DR

This paper proves that all graphs with girth at least 11 and no longer odd holes are 3-colorable, extending previous results for smaller girth values and supporting conjectures about broader classes.

Contribution

It establishes 3-colorability for graphs in igcup_{ ext{ell} extgreater=5} ext{G}_ ext{ell}, advancing understanding of coloring properties in graphs with specific girth and hole restrictions.

Findings

Graphs in igcup_{ ext{ell} extgreater=5} ext{G}_ ext{ell} are 3-colorable.

Extends previous results from ext{G}_2 and ext{G}_3.

Supports conjectures on coloring of graphs with large girth and restricted odd holes.

Abstract

For a number $\ell\geq 2$, let $\mathcal{G}_{\ell}$ denote the family of graphs which have girth $2\ell+1$ and have no odd hole with length greater than $2\ell+1$. Plummer and Zha conjectured that every 3-connected and internally 4-connected graph in $\mathcal{G}_2$ is 3-colorable. Wu, Xu, and Xu conjectured that every graph in $\bigcup_{\ell\geq2}\mathcal{G}_{\ell}$ is 3-colorable. Chudnovsky et al. and Wu et al., respectively, proved that every graph in $\mathcal{G}_2$ and $\mathcal{G}_3$ is 3-colorable. In this paper, we prove that every graph in $\bigcup_{\ell\geq5}\mathcal{G}_{\ell}$ is 3-colorable.