Measurable Brooks's Theorem for Directed Graphs

TL;DR

This paper extends Brooks's theorem to Borel directed graphs, establishing measurable and Baire-measurable d-colorings under certain conditions, and also proves a definable version of Gallai's theorem for list colorings.

Contribution

It provides a measurable version of Brooks's theorem and a definable Gallai's theorem for directed graphs with bounded degree.

Findings

Existence of measurable d-dicoloring unless the graph contains a complete symmetric directed graph on d+1 vertices.

Existence of Baire-measurable d-dicoloring under compatible Polish topology.

Borel degree-list-dicoloring for directed graphs with non-Gallai tree components.

Abstract

We prove a descriptive version of Brooks's theorem for directed graphs. In particular, we show that, if $D$ is a Borel directed graph on a standard Borel space $X$ such that the maximum degree of each vertex is at most $d \geq 3$, then unless $D$ contains the complete symmetric directed graph on $d + 1$ vertices, $D$ admits a $\mu$-measurable $d$-dicoloring with respect to any Borel probability measure $\mu$ on $X$, and $D$ admits a $\tau$-Baire-measurable $d$-dicoloring with respect to any Polish topology $\tau$ compatible with the Borel structure on $X$. We also prove a definable version of Gallai's theorem on list dicolorings for directed graphs by showing that any Borel directed graph of bounded degree whose connected components are not Gallai trees is Borel degree-list-dicolorable.