### Abstract

Given graphs G and H, an H-decomposition of G is a partition of the edge set of G such that each part is either a single edge or forms a graph isomorphic to H. Let ϕH(n) be the smallest number ϕ such that any graph G of order n admits an H-decomposition with at most ϕ parts.Here we determine the asymptotic of ϕH(n) for any fixed graph H as n tends to infinity.The exact computation of ϕH(n) for an arbitrary H is still an open problem. Bollobás [B. Bollobás, On complete subgraphs of different orders, Math. Proc. Cambridge Philos. Soc. 79 (1976) 19-24] accomplished this task for cliques. When H is bipartite, we determine ϕH(n) with a constant additive error and provide an algorithm returning the exact value with running time polynomial in logn.

Original language | Unknown |
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Pages (from-to) | 1041--1055 |

Journal | Journal of Combinatorial Theory, B |

Volume | 97 |

Issue number | 6 |

DOIs | |

Publication status | Published - 1 Jan 2007 |

## Cite this

Sousa, T. M. J. (2007). Minimum H-decompositions of graphs.

*Journal of Combinatorial Theory, B*,*97*(6), 1041--1055. https://doi.org/10.1016/j.jctb.2007.03.002