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

For a number $\ell \ge 2$, let $G_{\ell }$ denote the family of graphs which have girth $2\ell +1$ and have no odd hole with length greater than $2\ell +1$. Wu et al. conjectured that every graph in ${\bigcup }_{\ell \ge 2}{G}_{\ell }$ is 3-colorable. Chudnovsky et al. and Wu et al., respectively, proved that every graph in $G_2$ and $G_3$ is 3-colorable. In this paper, we prove that every graph in ${\bigcup }_{\ell \ge 5}{G}_{\ell }$ is 3-colorable.