Periodic colorings and orientations in infinite graphs

TL;DR

This paper investigates the existence of periodic colorings and orientations in infinite, locally finite graphs, establishing conditions under which such periodic structures exist or do not exist, with connections to symbolic dynamics and distributed computing.

Contribution

It characterizes when quasi-transitive graphs admit periodic colorings or orientations, providing new examples and conditions related to graph properties like treewidth and pathwidth.

Findings

Quasi-transitive graphs of bounded pathwidth have periodic proper colorings and orientations.

Certain Cayley graphs lack periodic orientations or non-trivial colorings.

Examples of quasi-transitive graphs with specific properties without periodic structures.

Abstract

We study the existence of periodic colorings and orientations in locally finite graphs. A coloring or orientation of a graph $G$ is periodic if the resulting colored or oriented graph is quasi-transitive, meaning that $V(G)$ has finitely many orbits under the action of the group of automorphisms of $G$ preserving the coloring or the orientation. When such a periodic coloring or orientation of $G$ exists, $G$ itself must be quasi-transitive and it is natural to investigate when quasi-transitive graphs have such periodic colorings or orientations. We provide examples of Cayley graphs with no periodic orientation or non-trivial coloring, and examples of quasi-transitive graphs of treewidth 2 without periodic orientation or proper coloring. On the other hand we show that every quasi-transitive graph $G$ of bounded pathwidth has a periodic proper coloring with $\chi(G)$ colors and a periodic orientation. We relate these problems with techniques and questions from symbolic dynamics and distributed computing and conclude with a number of open problems.