Continuous Edge Chromatic Numbers of Abelian Group Actions

TL;DR

This paper establishes the exact value of the continuous edge chromatic number for Schreier graphs arising from abelian group actions, showing it equals the classical chromatic number plus one.

Contribution

It provides a precise determination of the continuous edge chromatic number for Schreier graphs of $bZ^n$ actions, extending classical graph coloring results to a continuous setting.

Findings

Continuous edge chromatic number equals classical chromatic number plus one.

For standard generators, the number is $2n+1$.

Results apply to Bernoulli shift actions of free abelian groups.

Abstract

We prove that for any generating set $S$ of $\mathbb {Z}^n$, the continuous edge chromatic number of the Schreier graph of the Bernoulli shift action $G=F(S,2^{\mathbb{Z}^n})$ is $\chi'_c(G)=\chi'(G)+1$. In particular, for the standard generating set, the continuous edge chromatic number of $F(2^{\mathbb {Z}^n})$ is $2n+1$.