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Báo cáo toán học: "Rainbow H-factors of complete s-uniform r-partite hypergraph"

Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: Rainbow H-factors of complete s-uniform r-partite hypergraphs. | Rainbow H-factors of complete s-uniform r-partite hypergraphs Ailian Chen School of Mathematical Sciences Xiamen University Xiamen Fujian361005 P. R. China elian1425@sina.com Fuji Zhang School of Mathematical Sciences Xiamen University Xiamen Fujian361005 P. R. China fjzhang@xmu.edu.cn Hao Li Laboratoire de Recherche en Informatique UMR 8623 C. N. R. S. -Universite de Paris-sud 91405-Orsay Cedex France li@lri.fr Submitted Jan 19 2008 Accepted Jul 2 2008 Published Jul 14 2008 Mathematics Subject Classihcations 05C35 05C70 05C15 Abstract We say a s-uniform r-partite hypergraph is complete if it has a vertex partition Vi V2 . Vrg of r classes and its hyperedge set consists of all the s-subsets of its vertex set which have at most one vertex in each vertex class. We denote the complete s-uniform r-partite hypergraph with k vertices in each vertex class by Ts r k . In this paper we prove that if h r and s are positive integers with 2 s r h then there exists a constant k k h r s so that if H is an s-uniform hypergraph with h vertices and chromatic number x H r then any proper edge coloring of Ts r k has a rainbow H-factor. Keywords H-factors Rainbow uniform hypergraphs. 1 Introduction A hypergraph is a pair V E where V is a set of elements called vertices and E is a set of non-empty subsets of V called hyperedges or edges. A hypergraph H is called The work was partially supported by NSFC grant 10671162 and NNSF of china 60373012 . THE ELECTRONIC JOURNAL OF COMBINATORICS 15 2008 N26 1 s-uniform or an s-hypergraph if every edge has cardinality s. A graph is just a 2-uniform hypergraph. We say a hypergraph is r-partite if it has a vertex partition V1 V2 . Vrg of r classes such that each hyperedge has at most one vertex in each vertex class and a s-uniform r-partite hypergraph is complete if it has a vertex partition V1 V2 . Vrg of r classes and its hyperedge set consists of all the s-subsets of its vertex set which have at most one vertex in each vertex class. We denote the