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

Abstract

In 1998 the second author proved that there is an $\epsilon >0$ such that every graph satisfies $\chi \le \lceil (1-\epsilon )(\Delta +1)+\epsilon \omega \rceil$. The first author recently proved that any graph satisfying $\omega >\frac{2}{3}(\Delta +1)$ 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 Lovász Local Lemma, and Talagrand's Inequality.

arXiv comment10 pages. arXiv admin note: text overlap with arXiv:0911.1741