Integer and fractional packing of families of graphs

Let F be a family of graphs. For a graph G, the F-packing number, denoted νF (G), is the maximum number of pairwise edge-disjoint elements of F in G. A function ψ from the set of elements of F in G to [0, 1] is a fractional F-packing of G if ∑ e∈H∈F ψ(H) ≤ 1 for each e ∈ E(G). The fractional F-packing number, denoted ν F (G), is defined to be the maximum value of ∑ H∈(G F) ψ(H) over all fractional F -packings ψ. Our main result is that ν F (G) − νF (G) = o(|V (G)|2). Furthermore, a set of νF(G) − o(|V (G)|2) edge-disjoint elements of F in G can be found in randomized polynomial time. For the special case F = {H0} we obtain a significantly simpler proof of a recent difficult result of Haxell and Rödl [8] that ν H0 (G) − νH0(G) = o(|V (G)|2).