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 x(G) colors). It is shown that r(K(1) + G, K(1) + nH) <= k(hn + s-1) + 1 for large n. where x(G) = k >= 2. In particular, ifs is odd or s is even and hn is odd, then r(K(1) + K(k)(s), K(1) + nH) = k(hn + s-1) + 1, where K(k)(s) is a complete k-partite graph with s vertices in each part, implying that K(1) + nH is not (K(1) + K(k)(s))-good. Moreover, r(K(1) + sK(2), K(1) + nH) = 2hn + 1 for large n. (C) 2010 Published by Elsevier Ltd