  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.
