The planar cubic Cayley graphs

TL;DR

This paper provides a comprehensive classification and explicit descriptions of all planar cubic Cayley graphs, revealing their structure, counterexamples to existing conjectures, and supporting computational analysis.

Contribution

It offers a complete description, explicit presentations, and embeddings for all planar cubic Cayley graphs, including new counterexamples and computational insights.

Findings

Complete classification of planar cubic Cayley graphs

Counterexamples to Mohar, Bonnington, and Watkins conjectures

Supports Droms' conjecture through computational analysis

Abstract

We obtain a complete description of the planar cubic Cayley graphs, providing an explicit presentation and embedding for each of them. This turns out to be a rich class, comprising several infinite families. We obtain counterexamples to conjectures of Mohar, Bonnington and Watkins. Our analysis makes the involved graphs accessible to computation, corroborating a conjecture of Droms.