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

TL;DR

This paper provides a unified proof that connected graphs with chromatic number equal to maximum degree, excluding the complement of C7, contain either a large clique or an odd hole with high-degree vertices.

Contribution

It offers a self-contained proof extending previous results, removing the need for the Strong Perfect Graph Theorem for certain cases.

Findings

Graphs with $ ext{chi}(G)= ext{Delta}(G)$ contain large cliques or specific odd holes

Excludes the case of the complement of C7 from the main result

Provides a uniform proof for all degrees without case-by-case analysis

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 $\Delta(G)=4, 5, 6$.