Toward Vu's conjecture

Abstract

In 2002, Vu conjectured that graphs of maximum degree $Δ$ and maximum codegree at most $ζΔ$ have chromatic number at most $(ζ+o(1))Δ$. Despite its importance, the conjecture has remained widely open. The only direct progress so far has been obtained in the ``dense regime,'' when $ζ$ is close to $1$, by Hurley, de Verclos, and Kang. In this paper we provide the first progress in the sparse regime $ζ\ll 1$, the case of primary interest to Vu. We show that there exists $ζ_0 > 0$ such that for all $ζ\in [\log^{-32}Δ,ζ_0]$, the following holds: if $G$ is a graph with maximum degree $Δ$ and maximum codegree at most $ζΔ$, then $χ(G) \leq (ζ^{1/32} + o(1))Δ$. We derive this from a more general result that assumes only that the common neighborhood of any $s$ vertices is bounded rather than the codegrees of pairs of vertices. Our more general result also extends to the list coloring setting, which is of independent interest.