A classification of orientably-regular embeddings of complete multipartite graphs

Let Km[n]Km[n] be the complete multipartite graph with mm parts, while each part contains nn vertices. The orientably-regular embeddings of complete graphs Km[1]Km[1] have been determined by Biggs (1971)  [1], James and Jones (1985)  [14]. During the past twenty years, several papers such as Du et al. (2007, 2010)  [8] and [9], Jones et al. (2007, 2008)  [16] and [17], Kwak and Kwon (2005, 2008)  [18] and [19] and Nedela et al. (1997, 2002)  [22] and [23] contributed to the orientably-regular embeddings of complete bipartite graphs K2[n]K2[n] and the final classification was given by Jones  [15] in 2010. Based on our former paper (Zhang and Du, 2012)  [24], this paper gives a complete classification of orientably-regular embeddings of graphs Km[n]Km[n] for the general cases m≥3m≥3 and n≥2n≥2.