Hypergraphs are systems of finite sets, being the most general structures in discrete mathematics and powerful tools in dealing with discrete systems. In general, a branch of mathematics is built on some axioms. Infor...Hypergraphs are systems of finite sets, being the most general structures in discrete mathematics and powerful tools in dealing with discrete systems. In general, a branch of mathematics is built on some axioms. Informational scientists introduced the acyclic axiom for hypergraphs. In this paper, we first list several results concerning acyclic hypergraphs, in order to show that Acyclic-Axioms constitute the foundation of acyclic hypergraph theory. Then we give the basic theorem which shows that the Cycle-Axiom covers the Acyclic-Axioms and constitutes the foundation of hypergraph theory.展开更多
In this paper, the path through which the cycle axiom of hypergraphs was discovered will be retraced. The long process of discovery will be described, in particular how acyclic hypergraphs originated from the study of...In this paper, the path through which the cycle axiom of hypergraphs was discovered will be retraced. The long process of discovery will be described, in particular how acyclic hypergraphs originated from the study of relational database schemes and how cycles of hypergraphs originated from the study of acyclic hypergraphs.展开更多
Hypergraphs are the most general structures in discrete mathematics. Acyclic hypergraphs have been proved very useful in relational databases. New systems of axioms for paths, connectivity and cycles of hypergraphs ar...Hypergraphs are the most general structures in discrete mathematics. Acyclic hypergraphs have been proved very useful in relational databases. New systems of axioms for paths, connectivity and cycles of hypergraphs are constructed. The systems suit the structure properties of relational databases. The concepts of pseudo cycles and essential cycles of hypergraphs are introduced. They are relative to each other. Whether a family of cycles of a hypergraph is dependent or independent is defined. An enumeration formula for the maximum number of independent essential cycles of a hypergraph is given.展开更多
In this paper, some new concepts for hypergraphs are introduced. Based on the previous results, we do further research on cycle structures of hypergraphs and construct a more strictly complete cycle structure system o...In this paper, some new concepts for hypergraphs are introduced. Based on the previous results, we do further research on cycle structures of hypergraphs and construct a more strictly complete cycle structure system of hypergraphs.展开更多
The Turan number of a k-uniform hypergraph H,denoted by exk(n;H),is the maximum number of edges in any k-uniform hypergraph F on n vertices which does not contain H as a subgraph.Let Cl(k)denote the family of all k-un...The Turan number of a k-uniform hypergraph H,denoted by exk(n;H),is the maximum number of edges in any k-uniform hypergraph F on n vertices which does not contain H as a subgraph.Let Cl(k)denote the family of all k-uniform minimal cycles of length l;S(l1,…,lr)denote the family of hypergraphs consisting of unions of r vertex disjoint minimal cycles of lengthl1,…lr,respectively,and Cl(k)denote a k-uniform linear cycle of length l.We determine precisely exk(n;S(l1,…,lr)and exk(n;Cl1(k),…,Cl1(k)for sufficiently large n.Our results extend recent results of Füredi and Jiang who determined the Turan numbers for single k-uniform minimal cycles and linear cycles.展开更多
文摘Hypergraphs are systems of finite sets, being the most general structures in discrete mathematics and powerful tools in dealing with discrete systems. In general, a branch of mathematics is built on some axioms. Informational scientists introduced the acyclic axiom for hypergraphs. In this paper, we first list several results concerning acyclic hypergraphs, in order to show that Acyclic-Axioms constitute the foundation of acyclic hypergraph theory. Then we give the basic theorem which shows that the Cycle-Axiom covers the Acyclic-Axioms and constitutes the foundation of hypergraph theory.
基金Supported by the National Natural Science Foundation of China (No.10671199)
文摘In this paper, the path through which the cycle axiom of hypergraphs was discovered will be retraced. The long process of discovery will be described, in particular how acyclic hypergraphs originated from the study of relational database schemes and how cycles of hypergraphs originated from the study of acyclic hypergraphs.
文摘Hypergraphs are the most general structures in discrete mathematics. Acyclic hypergraphs have been proved very useful in relational databases. New systems of axioms for paths, connectivity and cycles of hypergraphs are constructed. The systems suit the structure properties of relational databases. The concepts of pseudo cycles and essential cycles of hypergraphs are introduced. They are relative to each other. Whether a family of cycles of a hypergraph is dependent or independent is defined. An enumeration formula for the maximum number of independent essential cycles of a hypergraph is given.
基金Supported by the National Natural Science Foundation of China(No.11626036)Natural Science Foundation of Beijing(No.1174015)
文摘In this paper, some new concepts for hypergraphs are introduced. Based on the previous results, we do further research on cycle structures of hypergraphs and construct a more strictly complete cycle structure system of hypergraphs.
基金partially supported by the National Natural Science Foundation of China(Nos.12131013,11871034)partially supported by the National Natural Science Foundation of China(Nos.11922112,12161141006)the Natural Science Foundation of Tianjin(Nos.20JCZDJC00840,20JCJQJC00090)。
文摘The Turan number of a k-uniform hypergraph H,denoted by exk(n;H),is the maximum number of edges in any k-uniform hypergraph F on n vertices which does not contain H as a subgraph.Let Cl(k)denote the family of all k-uniform minimal cycles of length l;S(l1,…,lr)denote the family of hypergraphs consisting of unions of r vertex disjoint minimal cycles of lengthl1,…lr,respectively,and Cl(k)denote a k-uniform linear cycle of length l.We determine precisely exk(n;S(l1,…,lr)and exk(n;Cl1(k),…,Cl1(k)for sufficiently large n.Our results extend recent results of Füredi and Jiang who determined the Turan numbers for single k-uniform minimal cycles and linear cycles.
