## Abstract

Let F= {H_{1}, . . . ,H_{k}} be a family of graphs. A graph G is called totally F-decomposable if for every linear combination of the form α_{1} e(H_{1}) + ⋯ + α_{k}e(H _{k}) = e(G) where each α_{i} is a nonnegative integer, there is a colouring of the edges of G with α_{1} + ⋯ + α_{k} colours such that exactly α_{i} colour classes induce each a copy of H_{i}, for i = 1, . . . ,k. We prove that if F is any fixed nontrivial family of trees then log n/n is a sharp threshold function for the property that the random graph G(n,p) is totally F-decomposable. In particular, if H is a tree with more than one edge, then log n/n is a sharp threshold function for the property that G(n,p) contains ⌊e(G)/e(H) ⌋ edge-disjoint copies of H.

Original language | English |
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Pages (from-to) | 893-910 |

Number of pages | 18 |

Journal | Combinatorics Probability and Computing |

Volume | 13 |

Issue number | 6 |

DOIs | |

State | Published - Nov 2004 |

## ASJC Scopus subject areas

- Theoretical Computer Science
- Statistics and Probability
- Computational Theory and Mathematics
- Applied Mathematics