Classification of regular embeddings of hypercubes of odd dimension

By a regular embedding of a graph into an orientable surface we mean a 2cell embedding with the automorphism group acting regularly on arcs. In 1997 Nedela and Škoviera [Europ. J. Comb. 18, 807-823] presented a construction giving for each solution of the congruence e2 ≡ 1(mod n) a regular embedding Me of the hypercube Qn. It was conjectured that all regular embeddings of Qn can be constructed in this way. This paper gives a classification of regular embeddings of hypercubes Qn for n odd. The conjecture is confirmed affirmatively for every odd n.