Graphs with $\chi=\Delta$ have big cliques

Abstract

Brooks' Theorem states that if a graph has $\Delta\ge 3$ and $\omega \le \Delta$, then $\chi \le \Delta$. Borodin and Kostochka conjectured that if $\Delta\ge 9$ and $\omega\le \Delta-1$, then $\chi\le \Delta-1$. We show that if $\Delta\ge 13$ and $\omega \le \Delta-4$, then $\chi\le \Delta-1$. For a graph $G$, let $\mathcal{H}$ denote the subgraph of $G$ induced by vertices of degree $\Delta$. We also show that if $\omega\le \Delta-1$ and $\omega(\mathcal{H})\le \Delta-6$, then $\chi\le \Delta-1$.