## Abstract

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 = {H_{0}} 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.

Original language | English |
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Pages (from-to) | 110-118 |

Number of pages | 9 |

Journal | Random Structures and Algorithms |

Volume | 26 |

Issue number | 1-2 |

DOIs | |

State | Published - Jan 2005 |

## ASJC Scopus subject areas

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