Generalized Cayley maps and Hamiltonian maps of complete graphs

A cellular embedding of a connected graph GG is said to be Hamiltonian if every face of the embedding is bordered by a Hamiltonian cycle (a cycle containing all the vertices of GG) and it is an mm-gonal embedding if every face of the embedding has the same length mm. In this paper, we establish a theory of generalized Cayley maps, including a new extension of voltage graph techniques, to show that for each even nn there exists a Hamiltonian embedding of KnKn such that the embedding is a Cayley map and that there is no nn-gonal Cayley map of KnKn if n≥5n≥5 is a prime. In addition, we show that there is no Hamiltonian Cayley map of KnKn if n=pen=pe, pp an odd prime and e>1e>1.