Equitable Colorings of Borel Graphs

TL;DR

This paper extends equitable coloring results from finite graphs to infinite Borel graphs, establishing Borel and measurable equitable colorings under various conditions, including invariance and degree constraints.

Contribution

It introduces Borel and measurable equitable coloring theorems for infinite graphs, generalizing classical finite graph results to the Borel setting.

Findings

Borel proper $k$-colorings exist for aperiodic Borel graphs with degree $ riangle$ when $k extgreater riangle$

Measurable equitable $ riangle$-colorings exist for graphs with small average degree and no $( riangle+1)$-cliques

Extensions of Brooks's theorem for measurable and list coloring are established

Abstract

Hajnal and Szemer\'{e}di proved that if $G$ is a finite graph with maximum degree $\Delta$, then for every integer $k \geqslant \Delta+1$, $G$ has a proper coloring with $k$ colors in which every two color classes differ in size at most by $1$; such colorings are called equitable. We obtain an analog of this result for infinite graphs in the Borel setting. Specifically, we show that if $G$ is an aperiodic Borel graph of finite maximum degree $\Delta$, then for each $k \geqslant \Delta + 1$, $G$ has a Borel proper $k$-coloring in which every two color classes are related by an element of the Borel full semigroup of $G$. In particular, such colorings are equitable with respect to every $G$-invariant probability measure. We also establish a measurable version of a result of Kostochka and Nakprasit on equitable $\Delta$-colorings of graphs with small average degree. Namely, we prove that if $\Delta \geqslant 3$, $G$ does not contain a clique on $\Delta + 1$ vertices, and $\mu$ is an atomless $G$-invariant probability measure such that the average degree of $G$ with respect to $\mu$ is at most $\Delta/5$, then $G$ has a $\mu$-equitable $\Delta$-coloring. As steps towards the proof of this result, we establish measurable and list coloring extensions of a strengthening of Brooks's theorem due to Kostochka and Nakprasit.