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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.

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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.

Item Type: Journal Article
Academic Unit/Department: Mathematics, Computing and Technology > Mathematics and Statistics
Item ID: 22779
Depositing User: Mike Grannell
Date Deposited: 18 Aug 2010 13:21
Last Modified: 02 Dec 2010 21:02
URI: http://oro.open.ac.uk/id/eprint/22779
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