临界完全图Ramsey数

Complete Critical Ramsey Numbers

摘要:

设G和H是任意的图，Ramsey数r(G,H)定义为最小的正整数r，使得图Kr的任意红蓝二边着色或存在单色的红色子图G，或存在单色的蓝色子图H.临界星图Ramsey数r*(G,H)为最小的正整数n，使得图Kr-K1,r-1-n的任意红蓝二边着色或存在单色的红色子图G，或存在单色的蓝色子图H.在临界星图启发下，临界完全图Ramsey数rK(G,H)定义为最大的正整数n，使得图Kr-Kn的任意红蓝二边着色或存在单色的红色子图G或存在单色的蓝色子图H.这里r为Ramsey数r(G,H).确定了rK(W1,n,K3)和rK(Cn,K3)，其中W1,n=K1+Cn为轮.

Abstract:

For graphs G and H, Ramsey number r(G,H) is the smallest integer r such that every 2coloring of Kr contains either a red copy of G or a blue copy of H. Star critical Ramsey number r*(G,H) is the smallest integer n such that every 2coloring of KrK1,r-1-n contains either a red copy of G or a blue copy of H. Under the inspiration of star critical Ramsey number, complete critical Ramsey number rK(G,H) is the largest integer n such that every 2coloring of KrKn contains either a red copy of G or a blue copy of H. In this paper, rK(Wn,Ka) and rK(Cn,K3) are determined. Wn=K1+Cn-1 is a wheel of size n.