扇形图与匹配图的临界星图Ramsey数

Star-critical Ramsey Number of Fan-graph Versus Matching

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)=2 m,n≤m,这里Fn=K1+nK2是扇形图.

Let Kn LIK1,k be a graph obtained from K,, 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-xLIJ K1,k contains a red G or a blue H, where N is the Ramsey number r （ G, H）. Let Fn = K1 ＋ nK2 be a fangraph. All critical graphs of r （ Fn, mK2 ） are determined in this note. Also the star-critical Ramsey numbers r ＊ （Fn, inK2） = m＋1 for n〉m≥1 and r ＊ （Fn, mK2） =2m for n≤m are obtained by discussing these critical graphs.