Regular Hamiltonian embeddings of the complete bipartite graph $K_{n,n}$ in an orientable surface

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

Abstract

An embedding $M$ of a graph $G$ is said to be regular if and only if for every two triples $(v_1,e_1,f_1)$ and $(v_2,e_2,f_2)$, where $e_i$ is an edge incident with the vertex $v_i$ and the face $f_i$, there exists an automorphism of $M$ which maps $v_1$ to $v_2$, $e_1$ to $e_2$ and $f_1$ to $f_2$. We show that for $n\not\equiv 0$ (mod 8) there is, up to isomorphism, precisely one regular Hamiltonian embedding of $K_{n,n}$ in an orientable surface, and that for $n\equiv 0$ (mod 8) there are precisely two such embeddings. We give explicit constructions for these embeddings as lifts of spherical embeddings of dipoles.

Viewing alternatives

No digital document available to download for this item

Item Actions

Export

Recommendations