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 \(G \otimes K_2\), 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 \(K_n \otimes K_2\) (known as the cocktail-party graphs) and of the \(n\)-dipoles \(D_n\), the graphs consisting of two vertices and \(n\) parallel edges joining them. In the first case we show, in particular, that regular embeddings of \(K_n \otimes K_2\) exist only if \(n\) is a prime power \(p^l\), and there are \(2 \phi (n - 1)\) or \(\phi (n - 1)\) isomorphism classes of such maps (where \(\phi\) 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 \(D_n\) exist for each positive integer \(n\), and their number is a power of 2 depending on the decomposition of \(n\) into primes.
scientific article; zbMATH DE number 934814