On the ultimate independence ratio of a graph

We study the ultimate independence ratio I(G) of a graph G, defined as the limit, as k→∞, of the sequence of independence ratios of the powers Gk. We prove that I(H)≤I(G) if there is a homomorphism of G to H. This allows us to prove that 1/χ(G) ≤ I(G) ≤ 1/χƒ(G), where χ(G) and χƒ(G) are the chromatic and the fractional chromatic numbers of G, respectively. From this result we derive a number of consequences: we construct graphs with I(G) strictly between 1/χ(G) and i(G) (answering a question from an earlier paper). We estimate I(G) for some classes of graphs and, in some cases, we compute the exact value of I(G). In particular, we show that I(G) = 12 for all bipartite graphs G.