Orientable Vertex Primitive Complete Maps

An orientable vertex primitive complete map is a two-cell embedding of a complete graph into an orientable surface such that the automorphism group of this map acts primitively on its vertex set. The paper is devoted to the problem of enumerating orientable vertex primitive complete maps. For a given integer n, we derive the number of different such maps with n vertices. Furthermore, we obtain explicit formulas for the numbers of non-isomorphic orientable vertex primitive complete maps with n vertices.