Sparsity in Critical Graphs with Small Clique Number

Abstract

In 1998, Reed conjectured that for every graph $G$, $\chi(G) \leq \lceil \frac{1}{2}(\Delta(G)+1+\omega(G)) \rceil$, and proved that there exists $\varepsilon > 0$ such that $\chi(G) \leq \lceil (1 - \varepsilon)(\Delta(G)+1) + \varepsilon \omega(G) \rceil$ for every graph $G$. Recently, much effort has been made to prove this result for increasingly large values of $\varepsilon$ in graphs with sufficiently large maximum degree. One of the main lemmas used in deriving these bounds states that graphs which are list-critical are sparse. This result generally follows by applying a sufficient condition for list colouring complete multipartite graphs with parts of bounded size, and until recently a theorem of Erd\H{o}s, Rubin and Taylor for list colouring complete multipartite graphs with parts of size at most two was used. The current bottleneck in bounding $\chi(G)$ for an improved value of $\varepsilon$ is the case of small clique number. We derive new density lemmas exploiting this case by showing that our graph is contained in a complete multipartite graph with many parts of size three. In order to list colour in this setting, we apply a theorem of Noel, West, Wu and Zhu, as well as our own unbalanced variant of this result.