Star critical Ramsey Number of Fan graph Versus Matching

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.