Decomposition Of Graph And Hypergraph And Its Large Sets

Combination design is one of the important branches in Combinatorics.It is closely related to the disciplines of coding,cryptography,network communication theory and so on.In this paper,we mainly study the cycle decomposition problem of hypergraphs using t-design.On the basis of the known results of the cycle decomposition of the complete graph,the cycle decomposition of the complete bipartite 3-uniform hypergraph is discussed by decomposing the small-order hypergraphs to K4(3),C4(3),K4(3)-e required for the construction of the auxiliary design.First,the existence decomposing the λ-fold complete bipartite 3-uniform hypergraphs λKm,n(3)to K4(3)is studied.A K4(3)have two hyperedges of type{a,b,x} and two hyperedges of type {a,x,y}.So there is a decomposition of Km,n(3)to K4(3)only when m=n.Based on this,we further study the decomposition ofλKm,n(3)to K4(3).Using the conclusion of the complete graph’s factorization and the combination design,we get that there is a decomposition of Kn,n(3)to K4(3)only when n is even.And there is a decomposition of 2Kn,n(3)n to K4(3)only when n is odd.Further,the existence spectrum of decomposition λKm,n(3)to K4(3)is derived.Due to the particularity of its construction,the existence spectrum of the large set of the decompositions is also obtained.Second,we study the existence of the decomposition of λKm,n(3)to C4(3).The necessity of the decomposition of λKn,n(3)to C4(3)is n=0,1,2(mod 4).When n≡1(mod 4).The existence decomposing Kn,n(3)to C4(3)is proved according to the conclusion of the cycle decomposition of the complete bipartite graph.Based on K4(3)belonging to one type of C4(3),the existence spectrum of the decomposition Kn,n(3)to C4(3)is derived.Due to the necessary conditions of the decomposition Km,n(3)to C4(3)some existential conclusions are obtained.Last,the existence decomposing λKm,n(3)to K4(3)-e is studied.First,we study the existence spectrum of the decomposition λKn,n(3)to K4(3)-e.Because of its necessity 3| λn2(n-1)and n≥2,it is necessary to consider the existence decomposing Kn,n(3)to K4(3),-ewhen λ=1,n≡0,1,3(mod6)to K4(3)-e,and the existence decomposing Kn,n(3),to K4(3)-e when λ=3,n ≡ 2(mod 6).In this paper,the direction structure method is used to prove the existence decomposing the small-order complete bipartite 3-uniform hypergraphs to K4(3)-e required embedded design.And the existence spectrum decomposing λKn,n(3)to K4(3)-e is obtained by the 3-GDD of the combination design and the conclusion of BIBD.Further,using the same method,we study the situation of m≠n,and obtain the partial conclusions.