Borel Edge Colorings for Finite Dimensional Groups

TL;DR

This paper explores the use of Borel asymptotic dimension to improve Borel edge colorings of Schreier graphs, achieving classical bounds and exact chromatic numbers for certain group actions.

Contribution

It introduces the application of Borel asymptotic dimension to Borel edge colorings, recovering Vizing's bound and determining exact chromatic numbers for abelian group actions.

Findings

Recovered Vizing's bound in specific cases

Determined exact Borel edge chromatic number for free abelian group actions

Showed the utility of Borel asymptotic dimension in graph coloring problems

Abstract

We study the potential of Borel asymptotic dimension, a tool introduced recently in arXiv:2009.06721, to help produce Borel edge colorings of Schreier graphs generated by Borel group actions. We find that it allows us to recover the classical bound of Vizing in certain cases, and also use it to exactly determine the Borel edge chromatic number for free actions of abelian groups.