Toward Vu's conjecture

TL;DR

This paper advances understanding of Vu's conjecture by proving new bounds on the chromatic number of graphs with bounded maximum degree and codegree in the sparse regime, extending to list coloring.

Contribution

It provides the first progress in the sparse regime of Vu's conjecture, establishing bounds for graphs with low codegree and generalizing to list coloring.

Findings

Proves bounds on chromatic number for sparse graphs with bounded codegree.

Extends results to list coloring setting.

Introduces a more general neighborhood bound condition.

Abstract

In 2002, Vu conjectured that graphs of maximum degree $\Delta$ and maximum codegree at most $\zeta \Delta$ have chromatic number at most $(\zeta+o(1))\Delta$. Despite its importance, the conjecture has remained widely open. The only direct progress so far has been obtained in the ``dense regime,'' when $\zeta$ is close to $1$, by Hurley, de Verclos, and Kang. In this paper we provide the first progress in the sparse regime $\zeta \ll 1$, the case of primary interest to Vu. We show that there exists $\zeta_0 > 0$ such that for all $\zeta \in [\log^{-32}\Delta,\zeta_0]$, the following holds: if $G$ is a graph with maximum degree $\Delta$ and maximum codegree at most $\zeta \Delta$, then $\chi(G) \leq (\zeta^{1/32} + o(1))\Delta$. 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.