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

Rachel Galindo, Jessica McDonald, Songling Shan

We give a uniform and self-contained proof that if $G$ is a connected graph with $\ddot{I}‡(G)=\hat{I}”(G)$ and $G\ne \overline{C_7}$, then $G$ contains either $K_{\hat{I}”(G)}$ or an odd hole whe…