On triangle-free list assignments
Jakub Przyby{\l}o

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 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 provided that vertices sharing a common colour in their lists do not induce a triangle in , which encompasses all cases covered by Molloy's theorem. This was thus far known to be true for lists of size , as implies a more general result due to Amini and Reed. We also prove that lists of length are sufficient if one replaces the triangle by any with , pushing also slightly the multiplicative factor of from Bernshteyn's result down to . All bounds presented are also valid…
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Taxonomy
TopicsAdvanced Graph Theory Research · Limits and Structures in Graph Theory
