On the orientable regular embeddings of complete multipartite graphs

Abstract

Let Km[n] be the complete multipartite graph with m parts, while each part contains n vertices. The regular embeddings of complete graphs Km[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 K2[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 Km[n] for m≥3. We in fact give a reduction theorem for the general classification, namely, we show that if Km[n] has an orientable regular embedding M, then either m=p and n=pe for some prime p≥5 or m=3 and the normal subgroup Aut0+(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 Aut0+(M) is abelian. We hope that our reduction theorem might be the first necessary approach leading to the general classification.