On the fractional chromatic number and the lexicographic product of graphs

For graphs G and H let GH] be their lexicographic product and let f (G) = inff(GK n ])=n j n = 1; 2; : : :g be the fractional chromatic number of G. For n 1 set G n = fG j (GK n ]) = nn(G)g. Then lim n!1 G n = fG j f (G) = (G)g: Moreover, we prove that for any n 2 the class G n forms a proper subclass of G n?1. As a by-product we show that if G is a-extremal, vertex transitive graph on (G)(G) ? 1 vertices, then for any graph H we have (GH]) = (G)(H) ? b(H)==(G)c.