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

Abstract

For a number $l\ge 2$, let $G_l$ 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 ${\bigcup }_{l\ge 2}{G}_l$ is 3-colourable. Chudnovsky et al., Wu et al., and Chen showed that every graph in $G_2$, $G_3$ and ${\bigcup }_{l\ge 5}{G}_l$ is 3-colourable, respectively. In this paper, we prove that every graph in $G_4$ is 3-colourable. This confirms Wu, Xu and Xu's conjecture.