Colouring Graphs with Sparse Neighbourhoods: Bounds and Applications

TL;DR

This paper advances Reed's conjecture by proving it for a larger epsilon value, using a new technique to bound chromatic numbers in graphs with sparse neighborhoods, and also improves bounds on the strong chromatic index.

Contribution

It introduces a novel method to bound the chromatic number in graphs with sparse neighborhoods, significantly advancing Reed's conjecture and improving bounds on the strong chromatic index.

Findings

Reed's conjecture holds for epsilon ≤ 1/26 with large maximum degree.

New technique bounds chromatic number in graphs with limited neighborhood edges.

Improved bounds on the strong chromatic index: χ'_s(G) ≤ 1.835 Δ(G)^2.

Abstract

Let $G$ be a graph with chromatic number $\chi$, maximum degree $\Delta$ and clique number $\omega$. Reed's conjecture states that $\chi \leq \lceil (1-\varepsilon)(\Delta + 1) + \varepsilon\omega \rceil$ for all $\varepsilon \leq 1/2$. It was shown by King and Reed that, provided $\Delta$ is large enough, the conjecture holds for $\varepsilon \leq 1/130,000$. In this article, we show that the same statement holds for $\varepsilon \leq 1/26$, thus making a significant step towards Reed's conjecture. We derive this result from a general technique to bound the chromatic number of a graph where no vertex has many edges in its neighbourhood. Our improvements to this method also lead to improved bounds on the strong chromatic index of general graphs. We prove that $\chi'_s(G)\leq 1.835 \Delta(G)^2$ provided $\Delta(G)$ is large enough.