Graphs with girth 9 and without longer odd holes are 3-colorable

TL;DR

This paper proves that graphs with girth 9 and no longer odd holes are 3-colorable, confirming a conjecture that all such graphs in a certain family are 3-colorable.

Contribution

The paper proves Wu, Xu, and Xu's conjecture for the case l=4, establishing 3-colorability for graphs with girth 9 and no longer odd holes.

Findings

Graphs in ${ m{ extbf{G}}}_4$ are 3-colorable.

Confirms Wu, Xu, and Xu's conjecture for l=4.

Extends known results to a new family of graphs.

Abstract

For a number $l\geq 2$, let ${\cal{G}}_l$ denote the family of graphs which have girth $2l+1$ and have no odd hole with length greater than $2l+1$. Wu, Xu and Xu conjectured that every graph in $\bigcup_{l\geq 2} {\cal{G}}_{l}$ is $3$-colorable. Chudnovsky et al., Wu et al., and Chen showed that every graph in ${\cal{G}}_2$, ${\cal{G}}_3$ and $\bigcup_{l\geq 5} {\cal{G}}_{l}$ is $3$-colorable respectively. In this paper, we prove that every graph in ${\cal{G}}_4$ is $3$-colorable. This confirms Wu, Xu and Xu's conjecture.