A Note on the Fractional Chromatic Number and the Lexicographic Product of Graphs

For graphs G and H let G[H] be their lexicographic product and let χf (G) = inf{χ(G[Kn])/n | n = 1, 2,...} be the fractional chromatic number of G. For n ≥ 1 set Gn = {G | χ(G[Kn]) = nχ(G)}. Then limn→ ∞ Gn = {G | χf (G) = χ(G)}. Moreover, we prove that for any n ≥ 2 the class Gn forms a proper subclass of Gn−1. As a by-product we show that if G is a χ∗-extremal, vertex transitive graph on χ(G)α(G) − 1 vertices, then for any graph H we have χ(G[H]) = χ(G)χ(H) − bχ(H)/α(G)c