A classification of orientably-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 orientably-regular embeddings of complete graphs K m [ 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)  [8,9], Jones et al. (2007, 2008)  [16,17], Kwak and Kwon (2005, 2008)  [18,19] and Nedela et al. (1997, 2002)  [22,23] contributed to the orientably-regular embeddings of complete bipartite graphs K 2 [ 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 K m [ n ] for the general cases m ≥ 3 and n ≥ 2 .