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 u to v'. A semiregular element of a permutation group is a nonidentity element having all cycles of equal length in its cycle decomposition. It is shown that every distance-transitive graph admits a semiregular automorphism. (C) 2009 Elsevier Ltd. All rights reserved.