An improved procedure for colouring graphs of bounded local density

TL;DR

This paper introduces an improved method for coloring graphs with bounded local density, leading to advances in longstanding conjectures by providing tighter bounds on chromatic numbers and strong chromatic index for large-degree graphs.

Contribution

It presents a new bound for the chromatic number of graphs with bounded local density, improving existing results and advancing the Erdős–Nešetřil and Reed conjectures.

Findings

Strong chromatic index at most 1.772Δ² for large Δ

Chromatic number at most ⌈0.881(Δ+1)+0.119ω⌉ for large Δ

Method can be adapted for graphs with bounded codegree

Abstract

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 the reduction of colours achieved through 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 $\lceil 0.881(\Delta+1)+0.119\omega\rceil$ for any graph $G$ with clique number $\omega$ and sufficiently large maximum degree $\Delta$. Additionally, we show how our methods can be adapted under the additional assumption that the codegree is at most $(1-\sigma)\Delta$, and establish what may be considered first progress towards a conjecture of Vu.