Integer and fractional packing of families of graphs

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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 ∑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) - 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 simpler proof of a recent difficult result of Haxell and Rödl [Combinatorica 21 (2001), 13-38] that ν*H0(G) - νH0(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.

Original languageEnglish
Pages (from-to)110-118
Number of pages9
JournalRandom Structures and Algorithms
Issue number1-2
StatePublished - Jan 2005

ASJC Scopus subject areas

  • Software
  • General Mathematics
  • Computer Graphics and Computer-Aided Design
  • Applied Mathematics


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