The Johansson--Molloy Theorem for DP-Coloring

TL;DR

This paper simplifies Molloy's proof of a bound on chromatic number for triangle-free graphs and extends the result to DP-coloring, broadening its applicability in graph coloring theory.

Contribution

It provides a streamlined proof of Molloy's bound and extends it to the more general DP-coloring framework, avoiding complex entropy methods.

Findings

Simplified proof of Molloy's chromatic bound for triangle-free graphs.

Extension of the bound to DP-coloring (correspondence coloring).

Avoidance of entropy compression method in the proof.

Abstract

The aim of this note is twofold. On the one hand, we present a streamlined version of Molloy's new proof of the bound $\chi(G) \leq (1+o(1))\Delta(G)/\ln \Delta(G)$ for triangle-free graphs $G$, avoiding the technicalities of the entropy compression method and only using the usual "lopsided" Lov\'asz Local Lemma (albeit in a somewhat unusual setting). On the other hand, we extend Molloy's result to DP-coloring (also known as correspondence coloring), a generalization of list coloring introduced recently by Dvo\v{r}\'ak and Postle.