Classification of Regular Embeddings of Complete Multipartite Graphs

A 2-cell embedding of a graph Gamma into a closed (orientable or nonorientable) surface is called regular if its automorphism group acts regularly on the flags. In this article, we classify the regular embeddings of the complete multipartite graph K-n,K-...,K-n. We show that if the number of partite sets is greater than 3, there exists no such embedding; and if the number of partite sets is 3, for any n, there exist one orientable regular embedding and one nonorientable regular embedding of K-n,K-n,K-n up to isomorphism. (C) 2017 Wiley Periodicals, Inc.