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

We give a uniform and self-contained proof that if $G$ is a connected graph with $χ(G) = Δ(G)$ and $G\neq \overline{C_7}$, then $G$ contains either $K_{Δ(G)}$ or an odd hole where every vertex has degree at least $Δ(G)-1$ 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 $Δ(G)=4, 5, 6$.