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

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.