Star-Critical Ramsey Numbers of Generalized Fans

For graphs F, G and H, let $$F\rightarrow (G,H)$$ signify that any red/blue edge coloring of F contains either a red G or a blue H. The Ramsey number R(G, H) is defined as the minimum r such that $$K_r\rightarrow (G,H)$$ , and the star-critical Ramsey number $$R_{{\mathbb {S}}}(G,H)$$ is defined as the maximum n such that $$K_r\setminus K_{1,n}\rightarrow (G,H)$$ , where $$r=R(G,H)$$ . We shall determine $$R_{{\mathbb {S}}}(K_2+G,K_1+nH)$$ to be $$v(H)n-\delta (H)-1$$ for all large n.