Regular Embeddings of Canonical Double Coverings of Graphs

This paper addresses the question of determining, for a given graphG, all regular maps havingGas their underlying graph, i.e., all embeddings ofGin closed surfaces exhibiting the highest possible symmetry. We show that ifGsatisfies certain natural conditions, then all orientable regular embeddings of its canonical double covering, isomorphic to the tensor productG⊗K2, can be described in terms of regular embeddings ofG. 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 productsKn⊗K2(known as the cocktail-party graphs) and of then-dipolesDn, the graphs consisting of two vertices and n parallel edges joining them. In the first case we show, in particular, that regular embeddings ofKn⊗K2exist only ifnis a prime powerpl, and there are 2φ(n−1) orφ(n−1) isomorphism classes of such maps (whereφis Eulerʹs function) according to whetherlis even or odd. Forleven an interesting new infinite family of regular maps is discovered. In the second case, orientable regular embeddings ofDnexist for each positive integern, and their number is a power of 2 depending on the decomposition ofninto primes.