## Abstract

Let K be a graph on r vertices and let G = (V,E) be another graph on |V| = n vertices. Denote the set of all copies of K in G by K,. A non-negative real-valued function f: K → ℝ_{+} is called a fractional K-factor if Σ_{K.v∈K∈K}f(K) ≤ 1 for every v ∈ V and Σ_{K∈K}f(K) = n/r. For a non-empty graph K let d(K) = e(K)/v(K) and d^{(1)} (K) = e(K)/(v(K) - 1). We say that K is strictly K_{1}-balanced if for every proper subgraph K′ K, d^{(1)} (K′) < d^{(1)} (K). We say that K is imbalanced if it has a subgraph K′ such that d(K′) > d(K). Considering a random graph process G̃ on n vertices, we show that if K is strictly K _{1}-balanced, then with probability tending to 1 as n → ∞, at the first moment τ_{0} when every vertex is covered by a copy of K, the graph G̃_{τ0} has a fractional K-factor. This result is the best possible. As a consequence, if K is K_{1}-balanced, we derive the threshold probability function for a random graph to have a fractional K-factor. On the other hand, we show that if K is an imbalanced graph, then for asymptotically almost every graph process there is a gap between τ_{0} and the appearance of a fractional K-factor. We also introduce and apply a criteria for perfect fractional matchings in hypergraphs in terms of expansion properties.

Original language | English |
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Pages (from-to) | 441-463 |

Number of pages | 23 |

Journal | Random Structures and Algorithms |

Volume | 30 |

Issue number | 4 |

DOIs | |

State | Published - Jul 2007 |

Externally published | Yes |