Coloring of graphs without long odd holes

A {\em hole} is an induced cycle of length at least 4, a $k$-hole is a hole of length $k$, and an {\em odd hole} is a hole of odd length. Let $\mathrm{\ell }\ge 2$ be an integer. Let ${\mathcal{A}}_{\mathrm{\ell }}$ be the family of graphs of girth at least $2\mathrm{\ell }$ and having no odd holes of length at least $2\mathrm{\ell }+3$, let ${\mathcal{B}}_{\mathrm{\ell }}$ be the triangle-free graphs which have no 5-holes and no odd holes of length at least $2\mathrm{\ell }+3$, and let ${\mathcal{G}}_{\mathrm{\ell }}$ be the family of graphs of girth $2\mathrm{\ell }+1$ and have no odd hole of length at least $2\mathrm{\ell }+5$. Chudnovsky {\em et al.} \cite{CSS2016} proved that every graph in ${\mathcal{A}}_2$ is 58000-colorable, and every graph in ${\mathcal{B}}_{\mathrm{\ell }}$ is $(\mathrm{\ell }+1)4^{\mathrm{\ell }-1}$-colorable. Lan and liu \cite{LL2023} showed that for $\mathrm{\ell }\ge 3$, every graph in ${\mathcal{G}}_{\mathrm{\ell }}$ is 4-colorable. It is not known whether there exists a small constant $c$ such that graphs of ${\mathcal{G}}_2$ are $c$-colorable. In this paper, we show that every graph in ${\mathcal{G}}_2$ is 1456-colorable, and every graph in ${\mathcal{A}}_3$ is 4-colorable. We also show that every 7-hole free graph in ${\mathcal{B}}_{\mathrm{\ell }}$ is $(12\mathrm{\ell }+8)$-colorable.