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.