Ramsey Numbers of Books and Fans

Abstract:

For given graphs G and H, Ramsey number R(G,H) is the smallest positive integer N such that any red/blue edgecoloring of KN contains either a red copy of G or a blue copy of H. Denote by G+H the graph obtained from disjoint G and H by adding edges connecting G and H completely. Let Bm=K2+mK1 and Fn=K1+nK2. It is shown that R(Bm,Fn)=4n+1 for n≥max{2,3m-2}； and R(F2,K2,n)=2n+3 for n≥38.