A short proof that $\chi$ can be bounded $\epsilon$ away from $\Delta+1$ towards $\omega$

TL;DR

This paper provides a shorter, simpler proof that the chromatic number of a graph can be bounded away from +1 by a positive , utilizing recent results on maximum cliques and stable sets.

Contribution

The authors present a simplified proof of a known bound on the chromatic number, leveraging recent findings on stable sets intersecting maximum cliques.

Findings

The proof is shorter and more accessible.

It confirms the bound on +1 with a simpler approach.

Includes an appendix with detailed intermediate results.

Abstract

In 1998 the second author proved that there is an $\epsilon>0$ such that every graph satisfies $\chi \leq \lceil (1-\epsilon)(\Delta+1)+\epsilon\omega\rceil$. The first author recently proved that any graph satisfying $\omega > \frac 23(\Delta+1)$ contains a stable set intersecting every maximum clique. In this note we exploit the latter result to give a much shorter, simpler proof of the former. We include, as a certificate of simplicity, an appendix that proves all intermediate results with the exception of Hall's Theorem, Brooks' Theorem, the Lov\'asz Local Lemma, and Talagrand's Inequality.