New regular embeddings of n ‐cubes Q n

We investigate highly symmetrical embeddings of the n ‐dimensional cube Q n into orientable compact surfaces which we call regular embeddings of Q n . We derive some general results and construct some new families of regular embeddings of Q n . In particular, for n  = 1, 2,3(mod 4), we classify the regular embeddings of Q n which can be expressed as balanced Cayley maps. © 2004 Wiley Periodicals, Inc. J Graph Theory 46: 297–312, 2004