On the order of semiregular automorphisms of cubic vertex-transitive graphs

TL;DR

This paper proves a dichotomy for finite connected cubic vertex-transitive graphs, showing they either have a large semiregular automorphism or are bounded in size, advancing understanding of symmetry in such graphs.

Contribution

It establishes a new bound on the order of semiregular automorphisms in cubic vertex-transitive graphs, linking automorphism properties to graph size.

Findings

Existence of a semiregular automorphism of order at least 6 or bounded vertex count

Bounded size of graphs without large semiregular automorphisms

Enhanced understanding of automorphism structure in cubic vertex-transitive graphs

Abstract

We prove that, if $\Gamma$ is a finite connected cubic vertex-transitive graph, then either there exists a semiregular automorphism of $\Gamma$ of order at least $6$, or the number of vertices of $\Gamma$ is bounded above by an absolute constant.