A characterisation of all vertex-transitive finite graphs of connectivity < 5

TL;DR

This paper classifies all finite connected vertex-transitive graphs with connectivity less than 5, identifying them as either 5-connected or belonging to specific known classes, using Tutte-type decompositions.

Contribution

It provides a complete characterization of low-connectivity vertex-transitive graphs, expanding understanding of their structure and classification.

Findings

All such graphs are either 5-connected or belong to a finite list of classes.

The classification relies on Tutte-type canonical decompositions.

The work simplifies the understanding of vertex-transitive graphs with low connectivity.

Abstract

We characterise all vertex-transitive finite connected graphs as essentially 5-connected or on a short list of explicit graph-classes. Our proof heavily uses Tutte-type canonical decompositions.