The vertex-transitive TLF-planar graphs

TL;DR

This paper characterizes and enumerates all topologically locally finite planar vertex-transitive graphs, providing a finite representation and automaton-based approach, and determines when such graphs are Cayley graphs.

Contribution

It introduces a finite local representation and labeling scheme for TLF-planar vertex-transitive graphs, enabling enumeration and Cayley graph recognition.

Findings

All TLF-planar vertex-transitive graphs can be described by a finite automaton.

The paper provides a method to enumerate these graphs for any degree.

It offers a decision procedure to identify Cayley graphs among them.

Abstract

We consider the class of the topologically locally finite (in short TLF) planar vertex-transitive graphs, a class containing in particular all the one-ended planar Cayley graphs and the normal transitive tilings. We characterize these graphs with a finite local representation and a special kind of finite state automaton named labeling scheme. As a result, we are able to enumerate and describe all TLF-planar vertex-transitive graphs of any given degree. Also, we are able decide to whether any TLF-planar transitive graph is Cayley or not.