Optimal $\chi$-boundness of $\ell$-holed graphs

Abstract

A graph is {\em{$\ell$-holed}} if all of its induced cycles of length at least four have length exactly $\ell$. In the paper, we prove that if $G$ is an $\ell$-holed graph with odd $\ell\geq 7$, then $\chi(G)\leq {\lceil {\ell \over {\ell-1}}\omega(G) \rceil}$. This result is sharp.