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
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.
