Structural characterization of Cayley graphs

TL;DR

This paper characterizes Cayley graphs as a special class of vertex-transitive, deterministic, and simple graphs, providing a unified structural understanding under certain set-theoretic assumptions.

Contribution

It offers a new characterization of Cayley graphs using properties like vertex transitivity and cycle languages, extending their understanding beyond traditional algebraic definitions.

Findings

Cayley graphs coincide with rooted deterministic vertex-transitive simple graphs.

Cayley graphs are characterized by having all vertices share the same cycle language.

Under the axiom of choice, Cayley graphs are the deterministic, co-deterministic, vertex-transitive simple graphs.

Abstract

We show that the directed labelled Cayley graphs coincide with the rooted deterministic vertex-transitive simple graphs. The Cayley graphs are also the strongly connected deterministic simple graphs of which all vertices have the same cycle language, or just the same elementary cycle language. Under the assumption of the axiom of choice, we characterize the Cayley graphs for all group subsets as the deterministic, co-deterministic, vertex-transitive simple graphs.