An improved procedure for colouring graphs of bounded local density

We develop an improved bound for the chromatic number of graphs of maximum degree $\Delta$ under the assumption that the number of edges spanning any neighbourhood is at most $(1-\sigma)\binom{\Delta}{2}$ for some fixed $0<\sigma<1$. The leading term in this bound is best possible as $\sigma\to0$. As two consequences, we advance the state of the art in two longstanding and well-studied graph colouring conjectures, the Erd\H{o}s-Ne\v{s}et\v{r}il conjecture and Reed's conjecture. We prove that the strong chromatic index is at most $1.772\Delta^2$ for any graph $G$ with sufficiently large maximum degree $\Delta$. We prove that the chromatic number is at most $\lceil 0.801(\Delta+1)+0.199\omega\rceil$ for any graph $G$ with clique number $\omega$ and sufficiently large maximum degree $\Delta$.Comment: 21 page

Colouring Squares of Claw-free Graphs

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Colouring Squares of Claw-free Graphs

International audienceIs there some absolute ε > 0 such that for any claw-free graph G, the chromatic number of the square of G satisûes χ(G 2 ) ≤ (2 − ε)ω(G) 2 , where ω(G) is the clique number of G? Erdős and Nešetřil asked this question for the speciûc case where G is the line graph of a simple graph, and this was answered in the aõrmative by Molloy and Reed. We show that the answer to the more general question is also yes, and, moreover, that it essentially reduces to the original question of Erdős and Nešetřil

Colouring squares of claw-free graphs

Contains fulltext : 176374.pdf (preprint version ) (Open Access