摘要
Given two graphs G and H,the Ramsey number R(G,H)is the smallest positive integer N such that every 2-coloring of the edges of K_(N)contains either a red G or a blue H.Let K_(N-1)■K_(1,k)be the graph obtained from K_(N-1)by adding anew vertexνconnecting k vertices of K_(N-1).A graph G withχ(G)=k+1 is called edge-critical if G contains an edge e such thatχ(G-e)=k.A considerable amount of research has been conducted by previous scholars on Ramsey numbers ofgraphs.In this study,we show that for an edge-critical graph G with x(G)=k+1,when k≥2,1≥2,and n is sufficiently large,R(G,K_(1)+nK_(t))=knt+1 and r,(G,K_(1)+nK_(t))=(k-1)nt+1.
给定两个图G和H,拉姆齐数R(G,H)是最小的正整数N,使得对K_(N)的边的任意一种2染色图中或包含一个红色图G或包含一个蓝色图H。记K_(N)■K_(1,k)是通过在K_(N-1)外添加一个顶点并连接K_(N-1个)的k顶点所得到的图。对一个色数为k+1的图G,如果它含一条边使得χ(G-e)=k,则称G为边临界图。前人对图拉姆齐数进行了大量研究。本文证明了对给定的色数为k+1的边临界图G,当k≥2,t≥2且n足够大时,R(G,K_(1)+nK_(t))=knt+1并且r,(G,K_(1)+nK_(t))=(k-1)nt+t。
作者
Taiping Jiang
Xinmin Hou
蒋太平;侯新民(中国科学技术大学数学科学学院,安徽合肥230026;中国科学技术大学中国科学院吴文俊数学重点实验室,安徽合肥230026;中国科学技术大学合肥国家实验室,安徽合肥230088)
基金
supported by the National Key Research and Development Program of China(2023YFA1010200,2020YFA0713100)
the National Natural Science Foundation of China(12071453)
the Innovation Program for Quantum Science and Technology(2021ZD0302902).
作者简介
Taiping Jiang received his M.D.degree from the University of Science and Technology of China in 2024.His research mainly focuses on graph and conbinatorics;通讯作者:侯新民,E-mail:xmhou@ustc.edu.cn,is a Professor at the University of Science and Technology of China.He received his Ph.D.degree in Mathematics from Dalian University of Technology in 2002.His research mainly focuses on extreme graph theory,structural graph theory,and combinatorial optimization theory.
