Classification of regular embeddings of n-dimensional cubes

An orientably-regular map is a 2-cell embedding of a connected graph or multigraph into an orientable surface, such that the group of all orientation-preserving automorphisms of the embedding has a single orbit on the set of all arcs (incident vertex-edge pairs). Such embeddings of the n-dimensional cubes Q(n) were classified for all odd n by Du, Kwak and Nedela in 2005, and in 2007, Jing Xu proved that for n = 2m where in is odd, they are precisely the embeddings constructed by Kwon in 2004. Here, we give a classification of orientably-regular embeddings of Q(n) for all n. In particular, we show that for all even n (= 2m), these embeddings are in one-to-one correspondence with elements sigma of order 1 or 2 in the symmetric group S-n such that sigma fixes n, preserves the set of all pairs B-i = {i, i + m} for 1 <= i <= m, and induces the same permutation on this set as the permutation B-i bar right arrow B-f(i) for some additive bijection f : Z(m) -> Z(m). We also give formulae for the numbers of embeddings that are reflexible and chiral, respectively, showing that the ratio of reflexible to chiral embeddings tends to zero for large even n.