Regular embeddings of Kn, n where n is a power of 2. I: Metacyclic case

A 2-cell embedding of a graph in an orientable closed surface is called regular if its automorphism group acts regularly on arcs of the embedded graph. The aim of this and of the associated consecutive paper is to give a classification of regular embeddings of complete bipartite graphs Kn,n, where n = 2. The method involves groups G which factorise as a product XY of two cyclic groups of order n so that the two cyclic factors are transposed by an involutory automorphism. In particular, we give a classification of such groups G. Employing the classification we investigate automorphisms of these groups, resulting in a classification of regular embeddings of Kn,n based on that for G. We prove that given n = 2, e ≥ 3 there are, up to map isomorphism, exactly 2e−2+4 regular embeddings of Kn,n. Our analysis splits naturally into two cases depending on whether the group G is metacyclic or not.