Cliques and High Odd Holes in Graphs with Chromatic Number Equal to Maximum Degree

Abstract

We give a uniform and self-contained proof that if $G$ is a connected graph with $\chi(G) = \Delta(G)$ and $G\neq \overline{C_7}$, then $G$ contains either $K_{\Delta(G)}$ or an odd hole where every vertex has degree at least $\Delta(G)-1$ in $G$. This was previously proved in series of two papers by Chen, Lan, Lin, and Zhou, who used the Strong Perfect Graph Theorem for the cases $\Delta(G)=4, 5, 6$.