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An improved procedure for colouring graphs of bounded local density

We develop an improved bound for the chromatic number of graphs of maximum degree $Δ$ under the assumption that the number of edges spanning any neighbourhood is at most $(1-σ)\binomΔ{2}$ for some fixed $0<σ<1$. The leading term in the reduction of colours achieved through this bound is best possible as $σ\to0$. As two consequences, we advance the state of the art in two longstanding and well-studied graph colouring conjectures, the Erdős-Nešetřil conjecture and Reed&#39;s conjecture. We prove that the strong chromatic index is at most $1.772Δ^2$ for any graph $G$ with sufficiently large maximum degree $Δ$. We prove that the chromatic number is at most $\lceil 0.881(Δ+1)+0.119ω\rceil$ for any graph $G$ with clique number $ω$ and sufficiently large maximum degree $Δ$. Additionally, we show how our methods can be adapted under the additional assumption that the codegree is at most $(1-σ)Δ$, and establish what may be considered first progress towards a conjecture of Vu.