Regular Embeddings of Canonical Double Coverings of Graphs

This paper addresses the question of determining, for a given graph G, all regular maps having G as their underlying graph, i.e., all embeddings of G in closed surfaces exhibiting the highest possible symmetry. We show that if G satisfies certain natural conditions, then all orientable regular embeddings of its canonical double covering, isomorphic to the tensor product G K2 , can be described in terms of regular embeddings of G. This allows us to ``lift'' the classification of regular embeddings of a given graph to a similar classification for its canonical double covering and to establish various properties of the ``derived'' maps by employing those of the ``base'' maps. We apply these results to determining all orientable regular embeddings of the tensor products Kn K2 (known as the cocktail-party graphs) and of the n-dipoles Dn , the graphs consisting of two vertices and n parallel edges joining them. In the first case we show, in particular, that regular embeddings of Kn K2 exist only if n is a prime power p, and there are 2,(n&1) or ,(n&1) isomorphism classes of such maps (where , is Euler's function) according to whether l is even or odd. For l even an interesting new infinite family of regular maps is discovered. In the second case, orientable regular embeddings of Dn exist for each positive integer n, and their number is a power of 2 depending on the decomposition of n into primes. 1996 Academic Press, Inc.