Colouring locally sparse graphs with the first moment method

TL;DR

This paper presents a new proof for an upper bound on the list chromatic number of locally sparse graphs using the first moment method, improving understanding of graph coloring in sparse conditions.

Contribution

It introduces a novel proof technique based on the first moment method for bounding the list chromatic number of graphs with local sparsity, strengthening previous results.

Findings

Bound on list chromatic number tight up to a factor of 2

Method provides asymptotically tight lower bounds on the number of colorings

Adapts Rosenfeld's counting argument for non-repetitive colorings

Abstract

We give a short proof of a bound on the list chromatic number of graphs $G$ of maximum degree $\Delta$ where each neighbourhood has density at most $d$, namely $\chi_\ell(G) \le (1+o(1)) \frac{\Delta}{\ln \frac{\Delta}{d+1}}$ as $\frac{\Delta}{d+1} \to \infty$. This bound is tight up to an asymptotic factor $2$, which is the best possible barring a breakthrough in Ramsey theory, and strengthens results due to Vu, and more recently Davies, P., Kang, and Sereni. Our proof relies on the first moment method, and adapts a clever counting argument developed by Rosenfeld in the context of non-repetitive colourings. As a final touch, we show that our method provides an asymptotically tight lower bound on the number of colourings of locally sparse graphs.