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Journal for Geometry and Graphics 7 (2003), No. 2, 173--190
Copyright Heldermann Verlag 2003

Non-orientable Maps and Hypermaps with Few Faces

Steve Wilson
Department of Mathematics and Statistics, Northern Arizona University, Flagstaff, AZ 86011, U.S.A.,

Antonio Breda d'Azevedo
Dep. de Matematica, Universidade de Aveiro, 3800 Aveiro, Portugal,

A map, or a cellular division of a compact surface, is often viewed as a cellular imbedding of a connected graph in a compact surface. It generalises to a hypermap by replacing "graph" with "hypergraph". In this paper we classify the non-orientable regular maps and hypermaps with size a power of 2, the non-orientable regular maps and hypermaps with 1, 2, 3, 5 faces and give a sufficient and necessary condition for the existence of regular hypermaps with 4 faces on non-orientable surfaces. For maps we classify the non-orientable regular maps with a prime number of faces. These results can be useful in classifications of non-orientable regular hypermaps or in non-existence of regular hypermaps in some non-orientable surface.

Keywords: Maps, hypermaps, graphs imbeddings, non-orientable surfaces.

MSC: 05C25; 05C30, 05C65, 05B45, 52C20, 57M07, 57M15, 57M50, 57M60.

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