Distance-transitive graphs admit semiregular automorphisms

A distance-transitive graph is a graph in which for every two ordered pairs of vertices ( u , v ) and ( u ′ , v ′ ) such that the distance between u and v is equal to the distance between u ′ and v ′ there exists an automorphism of the graph mapping u to u ′ and v to v ′ . A semiregular element of a permutation group is a non-identity element having all cycles of equal length in its cycle decomposition. It is shown that every distance-transitive graph admits a semiregular automorphism.