# Colouring triangle-free graphs with local list sizes

## TL;DR

This paper advances the understanding of coloring triangle-free graphs by providing refined bounds on their list, correspondence, and fractional chromatic numbers, considering local degree-based color availability, and introduces novel proof techniques involving the hard-core model.

## Contribution

It presents two new refinements of Molloy's result, allowing lower color availability for vertices with lower degree, and applies the hard-core model to fractional coloring.

## Key findings

- Refined bounds for list and correspondence coloring of triangle-free graphs.
- Application of the hard-core model to fractional coloring.
- Potential for broader use of the hard-core model in graph coloring proofs.

## 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 colour made available to vertices of lower degree to be accordingly lower. One result concerns list colouring and correspondence colouring, while the other concerns fractional colouring. 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.