The list chromatic number of graphs with small clique number

Abstract

The Lovasz Local Lemma, a cornerstone of the probabilistic method, is a powerful and widely used proof technique. In 2009, Moser introduced a technique called entropy compression to provide efficient algorithms which construct objects that the Local Lemma guarantees to exist. Recently, entropy compression has been used to develop more powerful versions of the Local Lemma which provide existence proofs in settings where the original Local Lemma does not apply. I will illustrate this technique with applications to graph colouring: (a) colouring triangle-free graphs, and (b) frugal colouring, where no colour can appear too many times in any neighbourhood.

We prove that every triangle-free graph with maximum degree $\D$ has list chromatic number at most $(1+o(1))\frac{\D}{\ln \D}$. This matches the best-known bound for graphs of girth at least 5. We also provide a new proof that for any $r\geq4$ every $K_r$-free graph has list-chromatic number at most $200r\frac{\D\ln\ln\D}{\ln\D}$.