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.
Categories: math.CO