Regular embeddings of complete multipartite graphs

In this paper, we classify all regular embeddings of the complete multipartite graphs Kp,…,p for a prime p into orientable surfaces. Also, the same work is done for the regular embeddings of the lexicographical product of any connected arc-transitive graph of prime order q with the complement of the complete graph of prime order p, where q and p are not necessarily distinct. Lots of regular maps found in this paper are Cayley maps.