Classification of nonorientable regular embeddings of Hamming graphs

By a regular embedding of a graph K in a surface we mean a 2-cell embedding of K in a compact connected surface such that the automorphism group acts regularly on flags. In this paper, we classify the nonorientable regular embeddings of the Hamming graph H ( d , n ) . We show that there exists such an embedding if and only if n = 2 and d = 2 , or n = 3 or 4 and d ≥ 1 , or n = 6 and d = 1 or 2 . We also give constructions and descriptions of these embeddings.