Cycle Decompositions Of Hypergraphs And Their Large Sets

Combinatorial design is an important branch in combinatorial mathematics.It is widely used in computer communication,network engineering and database theory.The language of graph decomposition can be used to describe combinatorial design.As a generalization of graph,it is of great significance to study the problem of hypergraph decomposition.In this paper,the problems of the tight 6-cycle decompositions of hypergraphs are studied by constructing the auxiliary designs.We discuss the tight 6-cycle decomposition of λ-fold complete bipartite 3-uniform hypergraphs λKn,n(3)and the tight 6-cycle decomposition of λ-fold complete 3-uniform hypergraphs λKn(3)by constructing the small orders required for the auxiliary designs.Secondly,we research the existence of large sets of decompositions of complete 4-partite 4-uniform hypergraphs K4×(2m)(4)is given by using the right coset theory of groups and complete automorphism groups.Firstly,we research the decomposition of a special kind of hypergraph,that is the decomposition of λ-fold complete bipartite 3-uniform hypergraph λKn,n(3)into tight 6cycles.The decomposition is denoted by Sλ(3,TC6(3),n,n).We obtain that the necessary condition for the existence of the decomposition is An2(n-1)≡ 0(mod 6)and n≥3.So,for the case λ=1,we only need to consider n≡0,3,4,7(mod 6);for the caseλ=3,we only need to consider n≡5,8(mod 6).Then,we give some recursive constructions and find the designs with small orders needed in recursive constructions.For the case where hypergraphs of large orders,they can be recursively generated by some designs with small orders.Further,the necessary conditions for the existence of S2(3,TC6(3),n,n)are also sufficient.So,an Sλ(3,TC6(3),n,n)exists if and only ifλn2(n-1)≡0(mod 6)and n≥3.Secondly,we research the decomposition of λ-fold complete 3-uniform hypergraph λKn(3)into tight 6-cycles,which is denoted by Sλ(3,TC6(3),n).We obtain that the necessary condition for the existence of the decomposition is An(n-1)(n-2)≡0(mod 36),λ(n-1)(n-2)≡0(mod 6)and n≥6.Then,we give a direct construction and find the designs with small orders needed in the direct construction.For the case where hypergraphs of large orders,they can be recursively generated by some designs with small orders.Further,the necessary conditions for the existence of Sλ(3,TC6(3),n)are also sufficient.So,an Sλ(3,TC6(3),n)exists if and only if λn(n-1)(n-2)≡0(mod 36),λ(n-1)(n-2)≡0(mod 6)and n≥6.In addition,we found applications of S(3,TC6(3),10)in the large sets of t-design.Finally,we discuss the problem of large sets of Hamilton cycle decompositions of complete 4-partite 4-uniform hypergraphs K4×(2m)(4).The existence spectrum of large sets of these decompositions is given by using the right coset theory of groups and complete automorphism groups.