Characterising planar Cayley graphs and Cayley complexes in terms of group presentations

TL;DR

This paper characterizes planar Cayley graphs and complexes through group presentations, providing conditions for planar embeddings and methods for their effective enumeration.

Contribution

It offers a new characterization of planar Cayley graphs via group presentations and describes how to effectively enumerate them.

Findings

Cayley graph can be embedded in the plane without accumulation points if and only if it is a 1-skeleton of a certain Cayley complex.

Provides a characterization of these graphs in terms of group presentations.

Shows that such Cayley graphs can be effectively enumerated.

Abstract

We prove that a Cayley graph can be embedded in the euclidean plane without accumulation points of vertices if and only if it is the 1-skeleton of a Cayley complex that can be embedded in the plane after removing redundant simplices. We also give a characterisation of these Cayley graphs in term of group presentations, and deduce that they can be effectively enumerated.