Regular Embeddings of Canonical Double Coverings of Graphs

Abstract

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 productsK_n⊗K₂(known as the cocktail-party graphs) and of then-dipolesD_n, the graphs consisting of two vertices and n parallel edges joining them. In the first case we show, in particular, that regular embeddings ofK_n⊗K₂exist only ifnis a prime powerp^l, and there are 2φ(n−1) orφ(n−1) isomorphism classes of such maps (whereφis Euler's 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 ofD_nexist for each positive integern, and their number is a power of 2 depending on the decomposition ofninto primes.