Presentations for Vertex Transitive Graphs

TL;DR

This paper introduces a generalized presentation method for vertex transitive graphs, proving key properties and providing new examples, including a non-Cayley 2-ended cubic vertex transitive graph.

Contribution

It extends Cayley graph constructions to encompass all vertex transitive graphs and answers longstanding questions about their structure.

Findings

Every countably infinite, connected, vertex transitive graph has a perfect matching.

Constructed a 2-ended cubic vertex transitive graph that is not a Cayley graph.

Generalized graph presentations to represent all vertex transitive graphs.

Abstract

We generalise the standard constructions of a Cayley graph in terms of a group presentation by allowing some vertices to obey different relators than others. The resulting notion of presentation allows us to represent every vertex transitive graph. As an intermediate step, we prove that every countably infinite, connected, vertex transitive graph has a perfect matching. Incidentally, we construct an example of a 2-ended cubic vertex transitive graph which is not a Cayley graph, answering a question of Watkins from 1990.