In [J. Combin. Theory Ser. B, 99 (2009), 447-454)], Li characterized the classification of vertex-transitive embeddings of complete graphs, and proposed how to enumerate such maps. In this paper, we study the counting problem of orientable vertex-transitive embeddings of ${\mathsf{K}}_p$, where $p\ge 5$ is a prime. Moreover, we obtain the number of non-isomorphic orientable vertex-transitive complete maps with $p$ vertices.

In the work by Li (J. Combin. Theory Ser. B, 99 (2009), 447–454.), the author characterized the classification of vertex transitive embeddings of complete graphs, and proposed how to enumerate such maps. In this paper, we study the counting problem of orientable vertex imprimitive complete maps, which is the automorphism group of this map acts imprimitively on its vertex set. Moreover, we obtain the number of non-isomorphic embeddings when the vertex-stabilizer subgroups of the automorphism groups of maps are isomorphic to ${\text{Z}}_{p-1}$ with odd prime $p$.