Measurable Brooks's Theorem for Directed Graphs
Cecelia Higgins
TL;DR
This work extends Brooks's and Gallai's theorems to descriptive settings for directed graphs by proving measurable and definable analogues for dicolorings under bounded degree assumptions. The authors establish a measurable Brooks-type theorem: for a Borel digraph $D=(X,A)$ with $\max d^{\max}(x)\le d$ and no symmetrized $K_{d+1}$, there exists a $\mu$-measurable $d$-dicoloring for any Borel probability measure $\mu$ and a $\tau$-Baire-measurable $d$-dicoloring for any compatible Polish topology $\tau$. They also obtain a definable Gallai-type result showing Borel degree-list-dicolorability on components not containing Gallai trees. The proofs integrate one-ended Borel functions and the Conley–Marks–Tucker-Drob spanning-forest framework, bridging descriptive set theory with directed-dicycle colorings and suggesting links to LOCAL algorithms and hypergraph colorings. Together, these results illuminate the landscape where descriptive/dynamic colorings of digraphs mirror, but distinct from, classical finite Brooks–Gallai theory, and open avenues for further exploration in algorithmic descriptive combinatorics.
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 $μ$-measurable $d$-dicoloring with respect to any Borel probability measure $μ$ on $X$, and $D$ admits a $τ$-Baire-measurable $d$-dicoloring with respect to any Polish topology $τ$ 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.