Integer and fractional packing of families of graphs

Let F be a family of graphs. For a graph G, the F-packing number, denoted ν_F(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 ∑_eεHεF ψ (H) ≤ 1 for each e ε E(G).The fractional F-packing number, denoted ν*_F(G), is defined to be the maximum value of ∑_Hε(G/F) ψ(H) overall fractional F-packings ψ. Our main result is that ν*_F(G) - ν_F(G) = o(|V(G)|²). Furthermore, a set of ν_F(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 simpler proof of a recent difficult result of Haxell and Rödl [Combinatorica 21 (2001), 13-38] that ν*_H0(G) - ν_H0(G) = o (|V(G)|²). Their result can be implemented in deterministic polynomial time. We also prove that the error term o(|V(G)|²) is asymptotically tight.