Coloring triangle-free graphs with local list sizes.

Ewan Davies,Ross J Kang,Rémi De Joannis De Verclos,François Pirot

doi:10.1002/rsa.20945

Abstract

We prove two distinct and natural refinements of a recent breakthrough result of Molloy (and a follow‐up work of Bernshteyn) on the (list) chromatic number of triangle‐free graphs. In both our results, we permit the amount of color made available to vertices of lower degree to be accordingly lower. One result concerns list coloring and correspondence coloring, while the other concerns fractional coloring. Our proof of the second illustrates the use of the hard‐core model to prove a Johansson‐type result, which may be of independent interest.

Highlights

The chromatic number of triangle-free graphs is a classic topic, cf. e.g. [16, 17], and has been deeply studied from many perspectives, including algebraic, probabilistic, and algorithmic
It is natural to ask what happens if fewer colours are supplied to vertices that are not of maximum degree; one might expect the low degree vertices to be easier to colour in a quantifiable way
Postle [4] initiated a modern and rather general treatment of this idea, including with respect to triangle-free graphs. (A conjecture of King [12] and related work are in the same vein.) We show the following result

Summary

Introduction

The chromatic number of triangle-free graphs is a classic topic, cf. e.g. [16, 17], and has been deeply studied from many perspectives, including algebraic, probabilistic, and algorithmic. We give a short and completely self-contained proof by analysing a probability distribution on independent sets known as the hard-core model in triangle-free graphs (Lemma 4), and demonstrating that to obtain the desired result it suffices to feed this distribution as input to a greedy fractional colouring algorithm (Lemma 3) Since it makes no use of the Lovasz Local Lemma, the proof is unlike any other derivation of a Johansson-type colouring result (regardless of local list sizes). P(Ai) ≤ xi (1 − xj), Aj ∈Di n the probability that no event in E occurs is at least (1 − xi) > 0

A fractional colouring algorithm

A local analysis of the hard-core model

Local fractional colouring

A list colouring lemma

Local list colouring

A necessary minimum degree condition for bipartite graphs