Semiregular elements in cubic vertex-transitive graphs and the restricted Burnside problem

TL;DR

This paper investigates the maximal order of semiregular automorphisms in cubic vertex-transitive graphs, disproving a conjecture in general but confirming it for Cayley and arc-transitive graphs using the restricted Burnside problem.

Contribution

The paper proves the non-unboundedness of semiregular automorphism orders in general cubic vertex-transitive graphs and confirms the conjecture for specific classes via the restricted Burnside problem.

Findings

Maximal order of semiregular elements does not tend to infinity in general.

Conjecture holds for Cayley and arc-transitive graphs.

Application of the restricted Burnside problem is key to the proof.

Abstract

In this paper, we prove that the maximal order of a semiregular element in the automorphism group of a cubic vertex-transitive graph X does not tend to infinity as the number of vertices of X tends to infinity. This gives a solution (in the negative) to a conjecture of Peter Cameron, John Sheehan and the author. However, with an application of the positive solution of the restricted Burnside problem, we show that this conjecture holds true when X is either a Cayley graph or an arc-transitive graph.