tailieunhanh - Báo cáo toán học: "Bipartite-uniform hypermaps on the sphere"

Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: Bipartite-uniform hypermaps on the sphere. | Bipartite-uniform hypermaps on the sphere Antonio Breda d Azevedo Rui Duarte Department of Mathematics University of Aveiro 3810-193 Aveiro Portugal breda@ Department of Mathematics University of Aveiro 3810-193 Aveiro Portugal rui@ Submitted Sep 29 2004 Accepted Dec 7 2006 Published Jan 3 2007 Mathematics Subject Classification 05C10 05C25 05C30 Abstract A hypermap is hypervertex- bipartite if its hypervertices can be 2-coloured in such a way that neighbouring hypervertices have different colours. It is bipartite-uniform if within each of the sets of hypervertices of the same colour hyperedges and hyperfaces all the elements have the same valency. The flags of a bipartite hypermap are naturally 2-coloured by assigning the colour of its adjacent hypervertices. A hypermap is bipartite-regular if the automorphism group acts transitively on each set of coloured flags. If the automorphism group acts transitively on the set of all flags the hypermap is regular. In this paper we classify the bipartite-uniform hypermaps on the sphere up to duality . Two constructions of bipartite-uniform hypermaps are given. All bipartite-uniform spherical hypermaps are shown to be constructed in this way. As a by-product we show that every bipartite-uniform hypermap H on the sphere is bipartite-regular. We also compute their irregularity group and index and also their closure cover Ha and covering core Ha. 1 Introduction A map generalises to a hypermap when we remove the requirement that an edge must join two vertices at most. A hypermap H can be regarded as a bipartite map where one of the two monochromatic sets of vertices represent the hypervertices and the other the hyperedges of H. In this perspective hypermaps are cellular embeddings of hypergraphs on compact connected surfaces two-dimensional compact connected manifolds without boundary in this paper we deal only with the boundary-free case. Research partially supported by R DU Matemática e Aplicacões of the .