Herzog,Hibi,Hreinddttir et al.introduced the class of closed graphs,and they proved that the binomial edge ideal JG of a graph G has quadratic GrSbner bases if G is closed.In this paper,we introduce the class of weakl...Herzog,Hibi,Hreinddttir et al.introduced the class of closed graphs,and they proved that the binomial edge ideal JG of a graph G has quadratic GrSbner bases if G is closed.In this paper,we introduce the class of weakly closed graphs as a generalization of the closed graph,and we prove that the quotient ring S/JG of the polynomial ring S=K[x1,...,xn,y1,...,yn]with K a field and n=|V(G)|is F-pure if G is weakly closed.This fact is a generalization of Ohtani's theorem.展开更多
Two definitions are given that Definitionl： an induced subgraph by a vertex vie G and its neighbors in G is defined as a vertex adjacent closed subgraph, and denoted by Qi （=G[V（Nvi）]）, with the vertex vi called ...Two definitions are given that Definitionl： an induced subgraph by a vertex vie G and its neighbors in G is defined as a vertex adjacent closed subgraph, and denoted by Qi （=G[V（Nvi）]）, with the vertex vi called the hub; and Definition2： A r（k,I）-I vertices graph is called the （k,l）-Ramsey graph, denoted by RG（k,1）, if RG（k,1） only contains cliques Kk.1 and the intersect QiNQj of any two nonadjacent vertices vi and vj of RG（k,I） contains only Kk-2. Meanwhile, the RG（k,l）＇s complement RG（I,k） contains only cliques Kl.l, and the intersect QiNQj of any two nonadjacent vertices vi and vj of RG（I,k） contains only Ki.2. On the basis of those definitions, two theorems are put forward and proved in this paper. They are Theoreml： the biggest clique in G is contained in some Qi of G, and Theorem 2： r（k,1） = [V（RG（k,I））I ＋ 1. With those definitions and theorems as well as analysis of chord property, a method for quick inspection and building RG（k,1） is proposed. Accordingly, RG（4,6） is built, it is a strongly 14-regular graph on order 35. We have tested RG（4,6） and its complement, as a result, they meet the defintion2, so we proclaim that r（4,6）=36.展开更多
By introducing the notions of L-spaces and L_r-spaces, a complete generalization of Kalton's closed graph theorem is obtained. It points out the class of L_r-spaces is the maximal class of range spaces for the clo...By introducing the notions of L-spaces and L_r-spaces, a complete generalization of Kalton's closed graph theorem is obtained. It points out the class of L_r-spaces is the maximal class of range spaces for the closed graph theorem when the class of domain spaces is the class of Mackey spaces with weakly * sequentially complete dual.Some examples are constructed showing that the class of L_r-spaces is strictly larger than the class of separable B_r-complete spaces.Some properties of L-spaces and L_r-spaces are discussed and the relations between B-complete (resp. B_r-complete) spaces and L-spaces (resp. L_r-spaces) are given.展开更多
On basis of two definitions that 1. an induced subgraph by a vertex vi E G and its neighbors in G is defined a vertex adjacent closed subgraph denoted by Qi （=G[V（Nvi）]）, with the vertex vi called the hub; 2. A r...On basis of two definitions that 1. an induced subgraph by a vertex vi E G and its neighbors in G is defined a vertex adjacent closed subgraph denoted by Qi （=G[V（Nvi）]）, with the vertex vi called the hub; 2. A r（k,1）-1 vertices connected graph is called a （k,l）-Ramsey graph denoted by RG（k,l）,if and only if 1. RG（k,l） contains only cliques of degree k-1, and its complement contains only cliques of degree l-l; 2. the intersect Qi∩Qj of any two nonadjacent vertices vi and vj of RG（k,1） contains Kk.2, and the intersect Qi∩Qj of any two nonadjacent vertices vi and vj of its complement RG（l,k） contains KI.2. Two theorems that theoreml ： the biggest clique in G is contained in some Qi of G, and theorem2： r（k,l）= [ V（RG（k,I）） I ＋1 are put forward and proved in this paper. With those definitions and theorems as well as analysis of property of chords a method for quick inspection and building RG（k,I） is proposed. Accordingly, RG（10,3） and its complement are built, which are respectively the strongly 29-regular graph and the strongly 10-regular graph on orders 40. We have tested RG（10,3） and its complement RG（3,10）,and gotten r（3,10）=41.展开更多
文摘Herzog,Hibi,Hreinddttir et al.introduced the class of closed graphs,and they proved that the binomial edge ideal JG of a graph G has quadratic GrSbner bases if G is closed.In this paper,we introduce the class of weakly closed graphs as a generalization of the closed graph,and we prove that the quotient ring S/JG of the polynomial ring S=K[x1,...,xn,y1,...,yn]with K a field and n=|V(G)|is F-pure if G is weakly closed.This fact is a generalization of Ohtani's theorem.
文摘Two definitions are given that Definitionl： an induced subgraph by a vertex vie G and its neighbors in G is defined as a vertex adjacent closed subgraph, and denoted by Qi （=G[V（Nvi）]）, with the vertex vi called the hub; and Definition2： A r（k,I）-I vertices graph is called the （k,l）-Ramsey graph, denoted by RG（k,1）, if RG（k,1） only contains cliques Kk.1 and the intersect QiNQj of any two nonadjacent vertices vi and vj of RG（k,I） contains only Kk-2. Meanwhile, the RG（k,l）＇s complement RG（I,k） contains only cliques Kl.l, and the intersect QiNQj of any two nonadjacent vertices vi and vj of RG（I,k） contains only Ki.2. On the basis of those definitions, two theorems are put forward and proved in this paper. They are Theoreml： the biggest clique in G is contained in some Qi of G, and Theorem 2： r（k,1） = [V（RG（k,I））I ＋ 1. With those definitions and theorems as well as analysis of chord property, a method for quick inspection and building RG（k,1） is proposed. Accordingly, RG（4,6） is built, it is a strongly 14-regular graph on order 35. We have tested RG（4,6） and its complement, as a result, they meet the defintion2, so we proclaim that r（4,6）=36.
文摘By introducing the notions of L-spaces and L_r-spaces, a complete generalization of Kalton's closed graph theorem is obtained. It points out the class of L_r-spaces is the maximal class of range spaces for the closed graph theorem when the class of domain spaces is the class of Mackey spaces with weakly * sequentially complete dual.Some examples are constructed showing that the class of L_r-spaces is strictly larger than the class of separable B_r-complete spaces.Some properties of L-spaces and L_r-spaces are discussed and the relations between B-complete (resp. B_r-complete) spaces and L-spaces (resp. L_r-spaces) are given.
文摘On basis of two definitions that 1. an induced subgraph by a vertex vi E G and its neighbors in G is defined a vertex adjacent closed subgraph denoted by Qi （=G[V（Nvi）]）, with the vertex vi called the hub; 2. A r（k,1）-1 vertices connected graph is called a （k,l）-Ramsey graph denoted by RG（k,l）,if and only if 1. RG（k,l） contains only cliques of degree k-1, and its complement contains only cliques of degree l-l; 2. the intersect Qi∩Qj of any two nonadjacent vertices vi and vj of RG（k,1） contains Kk.2, and the intersect Qi∩Qj of any two nonadjacent vertices vi and vj of its complement RG（l,k） contains KI.2. Two theorems that theoreml ： the biggest clique in G is contained in some Qi of G, and theorem2： r（k,l）= [ V（RG（k,I）） I ＋1 are put forward and proved in this paper. With those definitions and theorems as well as analysis of property of chords a method for quick inspection and building RG（k,I） is proposed. Accordingly, RG（10,3） and its complement are built, which are respectively the strongly 29-regular graph and the strongly 10-regular graph on orders 40. We have tested RG（10,3） and its complement RG（3,10）,and gotten r（3,10）=41.
