Ramsey goodness and generalized stars

Let G and H be fixed graphs with s(G)=s (the minimum number of vertices in a color class over all proper vertex-colorings of G with χ(G) colors). It is shown that r(K1+G,K1+nH)≤k(hn+s−1)+1 for large n, where χ(G)=k≥2. In particular, if s is odd or s is even and hn is odd, then r(K1+Kk(s),K1+nH)=k(hn+s−1)+1, where Kk(s) is a complete k-partite graph with s vertices in each part, implying that K1+nH is not (K1+Kk(s))-good. Moreover, r(K1+sK2,K1+nH)=2hn+1 for large n.