Symmetries of the Woolly Hat graphs

TL;DR

This paper investigates a specific class of tetravalent graphs called Woolly Hat graphs, proving their non-existence in the edge-transitive case and classifying those that are vertex-transitive.

Contribution

It introduces Woolly Hat graphs, proves their non-existence as edge-transitive graphs, and classifies all vertex-transitive Woolly Hat graphs.

Findings

No edge-transitive Woolly Hat graphs exist.

All vertex-transitive Woolly Hat graphs are classified.

The paper advances understanding of semiregular automorphisms in tetravalent graphs.

Abstract

A graph is edge-transitive if the natural action of its automorphism group on its edge set is transitive. An automorphism of a graph is semiregular if all of the orbits of the subgroup generated by this automorphism have the same length. While the tetravalent edge-transitive graphs admitting a semiregular automorphism with only one orbit are easy to determine, those that admit a semiregular automorphism with two orbits took a considerable effort and were finally classified in 2012. Of the several possible different ``types'' of potential tetravalent edge-transitive graphs admitting a semiregular automorphism with three orbits, only one ``type'' has thus far received no attention. In this paper we focus on this class of graphs, which we call the Woolly Hat graphs. We prove that there are in fact no edge-transitive Woolly Hat graphs and classify the vertex-transitive ones.