On the orientable regular embeddings of complete multipartite graphs

Let K m [ n ] be the complete multipartite graph with m parts, while each part contains n vertices. The regular embeddings of complete graphs K m [ 1 ] have been determined by Biggs (1971) [1], James and Jones (1985) [12] and Wilson (1989) [23]. During the past twenty years, several papers such as Du et al. (2007, 2010) [6,7], Jones et al. (2007, 2008) [14,15], Kwak and Kwon (2005, 2008) [16,17] and Nedela et al. (2002) [20] contributed to the regular embeddings of complete bipartite graphs K 2 [ n ] and the final classification was given by Jones [13] in 2010. Since then, the classification for general cases m ≥ 3 and n ≥ 2 has become an attractive topic in this area. In this paper, we deal with the orientable regular embeddings of K m [ n ] for m ≥ 3 . We in fact give a reduction theorem for the general classification, namely, we show that if K m [ n ] has an orientable regular embedding M , then either m = p and n = p e for some prime p ≥ 5 or m = 3 and the normal subgroup Aut 0 + ( M ) of Aut + ( M ) preserving each part setwise is a direct product of a 3-subgroup Q and an abelian 3 ′ -subgroup, where Q may be trivial. Moreover, we classify all the embeddings when m = 3 and Aut 0 + ( M ) is abelian. We hope that our reduction theorem might be the first necessary approach leading to the general classification.