Vertex transitive embeddings of complete graphs (Q1003847)

A map is a cellular embedding of a finite \(1\)-complex (graph) into a compact, connected, closed surface. An automorphism of a map is a permutation of the vertices that preserves the cell structure of the map. A map \({\mathcal M}\) is called vertex transitive if the automorphism group \(\text{Aut}{\mathcal M}\) acts transitively on the vertex set. The purpose of this paper is to give a characterization of vertex transitive embeddings of complete graphs \(K_n.\) The author shows that there are four types of such maps \({\mathcal M}\) and illustrates their existence by examples. For instance, one of the four conditions states that \({\mathcal M}\) is a sharp Cayley map of \(\text{Aut}{\mathcal M}.\) An open problem asks for the number of non-isomorphic vertex transitive maps with underlying graph \(K_n.\)