用r种颜色对图G的所有边着色,记着第i色的边构成的子图为Gi,如果存在一种着色方法使得对所有的1≤i≤r都满足Hi Gi,则称图G对于(H1,H2,…,Hr)可r着色.R am sey数R(H1,H2,…,Hr)是使得完全图Kn对于(H1,H2,…,Hr)不可r着色的最小正整数n....用r种颜色对图G的所有边着色,记着第i色的边构成的子图为Gi,如果存在一种着色方法使得对所有的1≤i≤r都满足Hi Gi,则称图G对于(H1,H2,…,Hr)可r着色.R am sey数R(H1,H2,…,Hr)是使得完全图Kn对于(H1,H2,…,Hr)不可r着色的最小正整数n.令m1>m2≥m3,E r.do.s等给出了当m1足够大时R(Cm1,Cm2,Cm3)的值.通过对m1不是足够大的情况进行研究,证明了当m≥5时,R(Cm,C3,C3)=5m-4;并给出了当m1≤7时R(Cm1,Cm2,Cm3)的值.展开更多
Existing critical point theories including metric and topological critical point theories are difficult to be applied directly to some concrete problems in particular polyhedral settings,because the notions of critica...Existing critical point theories including metric and topological critical point theories are difficult to be applied directly to some concrete problems in particular polyhedral settings,because the notions of critical sets could be either very vague or too large.To overcome these difficulties,we develop the critical point theory for nonsmooth but Lipschitzian functions defined on convex polyhedrons.This yields natural extensions of classical results in the critical point theory,such as the Liusternik-Schnirelmann multiplicity theorem.More importantly,eigenvectors for some eigenvalue problems involving graph 1-Laplacian coincide with critical points of the corresponding functions on polytopes,which indicates that the critical point theory proposed in the present paper can be applied to study the nonlinear spectral graph theory.展开更多
The Ramsey number is a foundational result in combinatorics. This article will introduce Ramsey number with the method of graph theory, and the Ramsey pricing theory is applied to the sales price and study of cross su...The Ramsey number is a foundational result in combinatorics. This article will introduce Ramsey number with the method of graph theory, and the Ramsey pricing theory is applied to the sales price and study of cross subsidy. Based on the status of our sales price and cross subsidy, Ramsey pricing methods theoretically guide adjustment thoughts of sales price and solve the practical problems in our life.展开更多
This is primarily an expository paper surveying up-to-date known results on the spectral theory of1-Laplacian on graphs and its applications to the Cheeger cut, maxcut and multi-cut problems. The structure of eigenspa...This is primarily an expository paper surveying up-to-date known results on the spectral theory of1-Laplacian on graphs and its applications to the Cheeger cut, maxcut and multi-cut problems. The structure of eigenspace, nodal domains, multiplicities of eigenvalues, and algorithms for graph cuts are collected.展开更多
Let K_(1,k)be a star of order k+1 and K_(n)■K_(1,k)the graph obtained from a complete graph K_(n)and an additional vertex v by joining v to k vertices of K_(n).For graphs G and H,the star-critical Ramsey number r_(*)...Let K_(1,k)be a star of order k+1 and K_(n)■K_(1,k)the graph obtained from a complete graph K_(n)and an additional vertex v by joining v to k vertices of K_(n).For graphs G and H,the star-critical Ramsey number r_(*)(G,H)is the minimum integer k such that any red/blue edge-coloring of K_(r-1)■K_(1,k)contains a red copy of G or a blue copy of H,where r is the classical Ramsey number R(G,H).Let C_(m)denote a cycle of order m and W_(n)a wheel of order n+1.Hook(2010)proved that r_(*)(W_(n),C_3)=n+3 for n≥6.In this paper,we show that r_(*)(W_(n),C_(m))=n+3 for m odd,m≥5 and n≥3(m-1)/2+2.展开更多
文摘用r种颜色对图G的所有边着色,记着第i色的边构成的子图为Gi,如果存在一种着色方法使得对所有的1≤i≤r都满足Hi Gi,则称图G对于(H1,H2,…,Hr)可r着色.R am sey数R(H1,H2,…,Hr)是使得完全图Kn对于(H1,H2,…,Hr)不可r着色的最小正整数n.令m1>m2≥m3,E r.do.s等给出了当m1足够大时R(Cm1,Cm2,Cm3)的值.通过对m1不是足够大的情况进行研究,证明了当m≥5时,R(Cm,C3,C3)=5m-4;并给出了当m1≤7时R(Cm1,Cm2,Cm3)的值.
基金supported by National Natural Science Foundation of China(Grant Nos.11822102 and 11421101)supported by Beijing Academy of Artificial Intelligence(BAAI)supported by the project funded by China Postdoctoral Science Foundation(Grant No.BX201700009)。
文摘Existing critical point theories including metric and topological critical point theories are difficult to be applied directly to some concrete problems in particular polyhedral settings,because the notions of critical sets could be either very vague or too large.To overcome these difficulties,we develop the critical point theory for nonsmooth but Lipschitzian functions defined on convex polyhedrons.This yields natural extensions of classical results in the critical point theory,such as the Liusternik-Schnirelmann multiplicity theorem.More importantly,eigenvectors for some eigenvalue problems involving graph 1-Laplacian coincide with critical points of the corresponding functions on polytopes,which indicates that the critical point theory proposed in the present paper can be applied to study the nonlinear spectral graph theory.
文摘The Ramsey number is a foundational result in combinatorics. This article will introduce Ramsey number with the method of graph theory, and the Ramsey pricing theory is applied to the sales price and study of cross subsidy. Based on the status of our sales price and cross subsidy, Ramsey pricing methods theoretically guide adjustment thoughts of sales price and solve the practical problems in our life.
基金supported by National Natural Science Foundation of China (Grant Nos. 11371038, 11471025, 11421101 and 61121002)
文摘This is primarily an expository paper surveying up-to-date known results on the spectral theory of1-Laplacian on graphs and its applications to the Cheeger cut, maxcut and multi-cut problems. The structure of eigenspace, nodal domains, multiplicities of eigenvalues, and algorithms for graph cuts are collected.
基金supported by the National Natural Science Foundation of China(Nos.11871270,12161141003,11931006)。
文摘Let K_(1,k)be a star of order k+1 and K_(n)■K_(1,k)the graph obtained from a complete graph K_(n)and an additional vertex v by joining v to k vertices of K_(n).For graphs G and H,the star-critical Ramsey number r_(*)(G,H)is the minimum integer k such that any red/blue edge-coloring of K_(r-1)■K_(1,k)contains a red copy of G or a blue copy of H,where r is the classical Ramsey number R(G,H).Let C_(m)denote a cycle of order m and W_(n)a wheel of order n+1.Hook(2010)proved that r_(*)(W_(n),C_3)=n+3 for n≥6.In this paper,we show that r_(*)(W_(n),C_(m))=n+3 for m odd,m≥5 and n≥3(m-1)/2+2.
