On the orientable regular embeddings of complete multipartite graphs

Abstract

Let $K_{m\left[n\right]}$ be the complete multipartite graph with $m$ parts, while each part contains $n$ vertices. The regular embeddings of complete graphs $K_{m\left[1\right]}$ 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 $K_{2\left[n\right]}$ and the final classification was given by Jones [13] in 2010. Since then, the classification for general cases $m\ge 3$ and $n\ge 2$ has become an attractive topic in this area. In this paper, we deal with the orientable regular embeddings of $K_{m\left[n\right]}$ for $m\ge 3$. We in fact give a reduction theorem for the general classification, namely, we show that if $K_{m\left[n\right]}$ has an orientable regular embedding $\mathcal{M}$, then either $m=p$ and $n=p^e$ for some prime $p\ge 5$ or $m=3$ and the normal subgroup ${\mathrm{Aut}}_0^{+}\left(\mathcal{M}\right)$ of ${\mathrm{Aut}}^{+}\left(\mathcal{M}\right)$ preserving each part setwise is a direct product of a 3-subgroup $Q$ and an abelian $3^{\prime }$-subgroup, where $Q$ may be trivial. Moreover, we classify all the embeddings when $m=3$ and ${\mathrm{Aut}}_0^{+}\left(\mathcal{M}\right)$ is abelian. We hope that our reduction theorem might be the first necessary approach leading to the general classification.