Classification of regular embeddings of hypercubes of odd dimension

By a regular embedding of a graph into a closed surface we mean a 2-cell embedding with the automorphism group acting regularly on flags. Recently, Kwon and Nedela [Non-existence of nonorientable regular embeddings of nn-dimensional cubes, Discrete Math., to appear] showed that no regular embeddings of the n  -dimensional cubes QnQn into nonorientable surfaces exist for any positive integer n>2n>2. In 1997, Nedela and Škoviera [Regular maps from voltage assignments and exponent groups, European J. Combin. 18 (1997) 807–823] presented a construction giving for each solution of the congruence e2≡1(modn) a regular embedding MeMe of the hypercube QnQn into an orientable surface. It was conjectured that all regular embeddings of QnQn into orientable surfaces can be constructed in this way. This paper gives a classification of regular embeddings of hypercubes QnQn into orientable surfaces for n odd, proving affirmatively the conjecture of Nedela and Škoviera for every odd n.