Brooks's theorem for measurable colorings

TL;DR

This paper extends Brooks's theorem to Borel graphs, showing that under certain conditions, such graphs admit measurable $d$-colorings, with new techniques involving spanning subforests and list coloring ideas.

Contribution

It introduces a novel method for constructing one-ended spanning subforests in Borel graphs and applies it to measurable colorings in group action graphs.

Findings

Borel graphs with degree ≤ d and no (d+1)-cliques admit measurable d-colorings.

New technique for constructing one-ended spanning subforests in Borel graphs.

Application to factor of IID d-colorings of Cayley graphs, with two exceptions.

Abstract

We generalize Brooks's theorem to show that if $G$ is a Borel graph on a standard Borel space $X$ of degree bounded by $d \geq 3$ which contains no $(d+1)$-cliques, then $G$ admits a $\mu$-measurable $d$-coloring with respect to any Borel probability measure $\mu$ on $X$, and a Baire measurable $d$-coloring with respect to any compatible Polish topology on $X$. The proof of this theorem uses a new technique for constructing one-ended spanning subforests of Borel graphs, as well as ideas from the study of list colorings. We apply the theorem to graphs arising from group actions to obtain factor of IID $d$-colorings of Cayley graphs of degree $d$, except in two exceptional cases.