Distance-transitive graphs admit semiregular automorphisms

A distance-transitive   graph is a graph in which for every two ordered pairs of vertices (u,v)(u,v) and (u′,v′)(u′,v′) such that the distance between uu and vv is equal to the distance between u′u′ and v′v′ there exists an automorphism of the graph mapping uu to u′u′ and vv to v′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.