Integer and fractional packing of families of graphs

• Authors: Raphael Yuster
• Venue: Random Structures & Algorithms - Proceedings of the Eleventh International Conference "Random Structures and Algorithms," August 9—13, 2003, Poznan, Poland
• Year: 2005

Let F be a family of graphs. For a graph G, the F-packing number, denoted &ngr;F(G), is the maximum number of pairwise edge-disjoint elements of F in G. A function &psgr; from the set of elements of F in G to [0, 1] is a fractional F-packing of G if σe∈H∈F &psgr;(H) ≤ 1 for each e ∈ E(G). The fractional F-packing number, denoted &ngr;F* (G), is defined to be the maximum value of σH∈( FG) &psgr;(H) over all fractional F-packings &psgr;. Our main result is that &ngr;F* (G)-&ngr;F(G) = o(|V(G)|2). Furthermore, a set of &ngr;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 &ngr;*H0 (G) - &ngr;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. © 2004 Wiley Periodicals, Inc. Random Struct. Alg., 2005