Non-orientable maps and hypermaps with few faces

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 nonorientable 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 nonorientable regular hypermaps or in non-existence of regular hypermaps in some non-orientable surface such as in [5].