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

TL;DR

This paper advances graph coloring theory by establishing new bounds on the chromatic number for graphs with specific degree and clique size conditions, extending previous conjectures and results.

Contribution

It proves that graphs with maximum degree at least 13 and certain clique constraints are colorable with fewer colors, improving bounds related to Borodin and Kostochka's conjecture.

Findings

Graphs with $ ext{max degree}\ge 13$ and $ ext{clique size}\le ext{max degree}-4$ are $( ext{max degree}-1)$-colorable.

Subgraph of vertices with maximum degree has limited clique size, leading to reduced chromatic number.

New bounds extend the range of graphs for which the chromatic number can be tightly controlled.

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 $\chi\le \Delta-1$.