Enumerating planar locally finite Cayley graphs

TL;DR

This paper characterizes and enumerates all planar locally finite Cayley graphs using finite automata called labeling schemes, enabling decision procedures for planarity in certain presentations.

Contribution

It introduces a finite representation of these graphs via labeling schemes and provides methods to enumerate and decide planarity for specific Cayley graphs.

Findings

Finite automata can represent all planar locally finite Cayley graphs.

Enumeration of such graphs is possible for a given degree.

Decidability of planarity is achieved for certain presentations.

Abstract

We characterize the set of planar locally finite Cayley graphs, and give a finite representation of these graphs by a special kind of finite state automata called labeling schemes. As a result, we are able to enumerate and describe all planar locally finite Cayley graphs of a given degree. This analysis allows us to solve the problem of decision of the locally finite planarity for a word-problem-decidable presentation. Keywords: vertex-transitive, Cayley graph, planar graph, tiling, labeling scheme