Non-existence of nonorientable regular embeddings of n-dimensional cubes

AbstractBy a regular embedding of a graph K into a surface we mean a two-cell embedding of K into a compact connected surface with the automorphism group acting regularly on flags. Regular embeddings of the n-dimensional cubes Qn into orientable surfaces exist for any positive integer n. In contrast to this, we prove the nonexistence of nonorientable regular embeddings of Qn for n>2