Decomposition Of Two Special Hypergraphs

In the past,graph theory has been widely discussed,and has great application in other mathematical fields.In order to solve more problems,it is necessary to study hypergraph.In this paper,we will use the method of combinatorial design to study hypergraph.In this paper,we study the hypergraph decomposition of two special types of hypergraphs,Triangular bipyramid and special tetrahedrons.The set of some t-uniform hypergraphs is denoted as ?,if a binary(X,B),X is a set of points which include v points,B is a family of hypergraphs,its vertex set is defined on some subsets of X.Every hypergraph in B is isomorphic to a hypergraph in ?,which is called a partition.And each t-subset of X is most(least)in ?partitions.Such a binary(X,B)is called t-(v,?,?)packing(covering).t-(v,?,?)packing(covering)have many situations,If the number of blocks can reach the maximum(minimum)value,it is called maximum packing(minimum covering).If every t-subsets of X happens to appear in ? blocks,such a binary(X,B)is called?-decomposition of hypergraph H,we also call it perfect packing or perfect covering,and we can also call this decomposition(H,?)-design.In this paper,we solve the decomposition problem of ?-bipartite 3-uniform hypergraph ?=(Triangular bipyramid),?=(special tetrahedron)and the decomposition problem of complete 3-uniform hypergraph ?=(special tetrahedron)when ?=1.The structure of the paper is as follows:In the first chapter,we give some definitions of hypergraph,the current research situation and achievements of hypergraph decomposition,and the research content of this paper.In the second chapter,we study the Triangular bipyramid decomposition of?Kn,n(3).It is proved the following theorem:except for two possible exceptional values S(3,TB,10,10)and S3(3,TB,14,14),the sufficient and necessary condition for the existence of S?(3,TB,n,n)is 6|?n2(n-1),2|An and n? 3.In Chapter 3,we study the special problem of special tetrahedron decomposition.It is proved that the necessary and sufficient condition for ?Kn,n(3)to be decomposed into a special tetrahedron is 4|An(n-1),n>3.In Chapter 4,we study the special tetrahedron decomposition of KV(3).It is proved that the necessary and sufficient condition for KV(3)to be decomposed into a special tetrahedron is v?1,2,6(mod 8)and v?6.