# Regular Hamiltonian embeddings of the complete bipartite graph in an orientable surface

Grannell, M. J.; Griggs, T. S. and Knor, M. (2003). Regular Hamiltonian embeddings of the complete bipartite graph in an orientable surface. Congressus Numerantium, 163, pp. 197–205.

## Abstract

An embedding of a graph is said to be regular if and only if for every two triples and , where is an edge incident with the vertex and the face , there exists an automorphism of which maps to , to and to . We show that for (mod 8) there is, up to isomorphism, precisely one regular Hamiltonian embedding of in an orientable surface, and that for (mod 8) there are precisely two such embeddings. We give explicit constructions for these embeddings as lifts of spherical embeddings of dipoles.