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

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.

18 pages, 1 figure

References in corpus (2)

• Asymptotic enumeration of vertex-transitive graphs of fixed valency
• Semiregular automorphisms of cubic vertex-transitive graphs

Cited by in corpus (5)

• Cubic vertex-transitive graphs of girth six
• On the order of semiregular automorphisms of cubic vertex-transitive graphs
• Semiregular automorphisms of cubic vertex-transitive graphs
• On orders of automorphisms of vertex-transitive graphs
• Finite cubic graphs admitting an cyclic group of automorphisms with at most three orbits on vertices