## Abstract

Our main result is that every graph G on n≥10^{4}r^{3} vertices with minimum degree δ(G)≥(1−1/10^{4}r^{3/2})n has a fractional K_{r}-decomposition. Combining this result with recent work of Barber, Kühn, Lo and Osthus leads to the best known minimum degree thresholds for exact (non-fractional) F-decompositions for a wide class of graphs F (including large cliques). For general k-uniform hypergraphs, we give a short argument which shows that there exists a constant c_{k}>0 such that every k-uniform hypergraph G on n vertices with minimum codegree at least (1−c_{k}/r^{2k−1})n has a fractional K_{r} ^{(k)}-decomposition, where K_{r} ^{(k)} is the complete k-uniform hypergraph on r vertices. (Related fractional decomposition results for triangles have been obtained by Dross and for hypergraph cliques by Dukes as well as Yuster.) All the above new results involve purely combinatorial arguments. In particular, this yields a combinatorial proof of Wilson's theorem that every large F-divisible complete graph has an F-decomposition.

Original language | English |
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Pages (from-to) | 148-186 |

Number of pages | 39 |

Journal | Journal of Combinatorial Theory. Series B |

Volume | 127 |

Early online date | 2 Jun 2017 |

DOIs | |

Publication status | Published - Nov 2017 |

## Keywords

- Cliques
- Fractional decomposition
- Minimum degree