扇形图与匹配图的临界星图Ramsey数

Star critical Ramsey Number of Fan graph Versus Matching

摘要:

对于完全图Kn和一个额外的顶点v，通过在v与Kn之间添加k条边所得出的图，记为KnK1,k.设G和H是任意的图，临界星图Ramsey数r*(G,H)定义为最小的正整数k，使得图KN-1K1,k的任意红蓝2边着色，或者存在单色的红色子图G，或者存在单色的蓝色子图H，这里N指的是Ramsey数r(G,H).文中找到了r(Fn,mK2)的所有临界图，利用这些临界图得到了临界星图Ramsey数r*(Fn,mK2)=m+1,n>m≥1,以及r*(Fn,mK2)=2m,n≤m，这里Fn=K1+nK2是扇形图.

Abstract:

Let KnK1,k be a graph obtained from Kn and an additional vertex v by joining v and k vertices of Kn. For graphs G and H, the star critical Ramsey number r*(G,H) is the smallest k such that every red/blue 2 edge coloring of KN-1K1,k contains a red G or a blue H, where N is the Ramsey number r(G,H). Let Fn=K1+nK2 be a fan graph. All critical graphs of r(Fn,mK2) are determined in this note. Also the star critical Ramsey numbers r*(Fn,mK2)=m+1 for n>m≥1 and r*(Fn,mK2)=2m for n≤m are obtained by discussing these critical graphs.