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 $ω$, $Δ$, and $χ$. 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 $ε> 0$) of relaxed bounds; the $ε= 0$ limit of which is Reed's upper bound.