Classification of regular embeddings of hypercubes of odd dimension

Abstract

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 n -dimensional cubes, Discrete Math., to appear] showed that no regular embeddings of the n-dimensional cubes Q n into nonorientable surfaces exist for any positive integer n > 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 e 2 ≡ 1 ( mod n ) a regular embedding M e of the hypercube Q n into an orientable surface. It was conjectured that all regular embeddings of Q n into orientable surfaces can be constructed in this way. This paper gives a classification of regular embeddings of hypercubes Q n into orientable surfaces for n odd, proving affirmatively the conjecture of Nedela and Škoviera for every odd n.