Integer and fractional packing of families of graphs

Let F be a family of graphs. For a graph G, the F-packing number, denoted nF(G), is the maximum number of pairwise edge-disjoint elements of F in G. A function p from the set of elements of F in G to [0, 1] is a fractional F-packing of G if σe∈H∈F p(H) ≤ 1 for each e ∈ E(G). The fractional F-packing number, denoted nFa (G), is defined to be the maximum value of σH∈( FG ) p(H) over all fractional F-packings p. Our main result is that nFa (G)-nF(G) = o(|V(G)|2). Furthermore, a set of nF(G)-o(|V(G)|2) edge-disjoint elements of F in G can be found in randomized polynomial time. For the special case F = lH0r we obtain a simpler proof of a recent difficult result of Haxell and Rodl [Combinatorica 21 (2001), 13–38] that naH0 (G) - nH0 (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. © 2004 Wiley Periodicals, Inc. Random Struct. Alg., 2005