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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.,
sew@odin.math.nau.edu
Antonio Breda d'Azevedo
Dep. de Matematica, Universidade de Aveiro, 3800 Aveiro, Portugal,
breda@mat.ua.pt
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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