The 4-cycle Decomposition Of Complete 3 Uniform Hypergraphs

Graph decomposition has a long history.It plays a key role in combina-torics,projective geometry,coding,information security and other fields.Hypergraph is the generalization of graph.As a hot issue in recent years,hypergraph decomposition has been widely used in many fields.For example,hypergraph decomposition theory can be applied to optimize the information transmission rate in secret sharing.It is one of the basic problems in the field of hypergraph research to study the decomposition of a complete k-uniform hypergraph.In 1960,Hanani first gave necessary and sufficient conditions for a k₄³decomposition of the complete 3-uniform hypergraph,which opened a new chapter in the study of hypergraph decomposition.In this paper,we consider the Berge cycle decomposition of complete 3-uniform hypergraph,especially,the case of 4-cycle decompositions.We use the theory of 3-designs to solve its existence problem.Actually,the k₄³that Hanani studied is a kind of strongly conditioned Berge 4-cycles.This thesis is organized as follows.In Chapter 1,the first section gives a brief introduction to the background,signifi-cance and research status of graph decompositions and hypergraph decompositions.The second section shows the main results of this thesis.Chapter 2 presents some basic concepts of combinatorial designs,and establishes the relationship between hypergraph decompositions and designs.It makes us to treat the problem of hypergraph decompositions from the point of view of design theory.The main recursive constructions used in this thesis are weighting and filling constructions.Chapter 3 first gives the classification of 3-uniform 4-cycles up to isomorphism.It is proved that,4-cycles composed of 4 points only have one type,4-cycles composed of five points have five types,4-cycles composed of six points have seven types,4-cycles composed of seven points have three types,and there are only one type of 4-cycles composed of eight points.For the 10 of the 13 unknown types of 4-cycles in the literature,we discuss them each in turn,and obtain the necessary and sufficient conditions for the complete 3-uniform hypergraph decomposition into 4-cycles with given types.In Chapter 4,we summarize this dissertation and give several problems for further research.