New upper bounds on the chromatic number of a graph

Abstract

We outline some ongoing work related to a conjecture of Reed \cite{reed97} on $\omega$, $\Delta$, and $\chi$. We conjecture that the complement of a counterexample $G$ to Reed's conjecture has connectivity on the order of $\log (|G|)$. We prove that this holds for a family (parameterized by $\epsilon >0$) of relaxed bounds; the $\epsilon =0$ limit of which is Reed's upper bound.