Generalized Cayley maps and Hamiltonian maps of complete graphs

A cellular embedding of a connected graph G is said to be Hamiltonian if every face of the embedding is bordered by a Hamiltonian cycle (a cycle containing all the vertices of G ) and it is an m -gonal embedding if every face of the embedding has the same length m . 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 n there exists a Hamiltonian embedding of K n such that the embedding is a Cayley map and that there is no n -gonal Cayley map of K n if n ≥ 5 is a prime. In addition, we show that there is no Hamiltonian Cayley map of K n if n = p e , p an odd prime and e > 1 .