On triangle-free list assignments

TL;DR

This paper extends Bernshteyn's proof techniques to show that triangle-free graphs and graphs with larger cliques are list-colorable with sizes close to Molloy's bounds, also applicable to correspondence coloring.

Contribution

It adapts Bernshteyn's proof to establish stronger list-coloring bounds for triangle-free and $K_r$-free graphs, improving previous results and generalizing to correspondence coloring.

Findings

List sizes of $(1+o(1))rac{ ext{max degree}}{ ext{log max degree}}$ suffice for triangle-free graphs.

List sizes of $2(r-2)rac{ ext{max degree} imes ext{log log max degree}}{ ext{log max degree}}$ suffice for $K_r$-free graphs.

Bounds are valid for correspondence coloring, broadening applicability.

Abstract

We show that Bernshteyn's proof of the breakthrough result of Molloy that triangle-free graphs are choosable from lists of size $(1+o(1))\Delta/\log\Delta$ can be adapted to yield a stronger result. In particular one may prove that such list sizes are sufficient to colour any graph of maximum degree $\Delta$ provided that vertices sharing a common colour in their lists do not induce a triangle in $G$, which encompasses all cases covered by Molloy's theorem. This was thus far known to be true for lists of size $(1000+o(1))\Delta/\log\Delta$, as implies a more general result due to Amini and Reed. We also prove that lists of length $2(r-2)\Delta \log_2\log_2\Delta/\log_2\Delta$ are sufficient if one replaces the triangle by any $K_r$ with $r\geq 4$, pushing also slightly the multiplicative factor of $200r$ from Bernshteyn's result down to $2(r-2)$. All bounds presented are also valid within the more general setting of correspondence colourings.