Regular hamiltonian embeddings of Kn, n and regular triangular embeddings of Kn, n, n

We give a group-theoretic proof of the following fact, proved initially by methods of topological design theory: Up to isomorphism, the number of regular hamiltonian embeddings of Kn,n is 2 or 1, depending on whether n is a multiple of 8 or not. We also show that for each n there is, up to isomorphism, a unique regular triangular embedding of Kn,n,n. This is a preprint of an article accepted for publication in Discrete Mathematics c ©2007 (copyright owner as specified in the journal).