## Abstract

A family of graphs possesses the common gcd property if the greatest common divisor of the degree sequence of each graph in the family is the same. In particular, any family of trees has the common gcd property. Let F = {H_{1},...,H_{r}} be a family of graphs having the common gcd property, and let d be the common gcd. It is proved that there exists a constant N = N(F) such that for every n > N for which d divides n -1, and for every equality of the form α_{1}e(H_{1}) +⋯+α_{r}e(H_{r})= (n/2), where α_{1},...,α_{r} are nonnegative integers, the complete graph K_{n} has a decomposition in which each H_{i} appears exactly α_{i} times. In case F is a family of trees the bound N(F) is shown to be polynomial in the size of F, and, furthermore, a polynomial (in n) time algorithm which generates the required decomposition is presented.

Original language | English |
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Pages (from-to) | 67-77 |

Number of pages | 11 |

Journal | Discrete Mathematics |

Volume | 243 |

Issue number | 1-3 |

DOIs | |

State | Published - 28 Jan 2002 |

## Keywords

- Decompositions
- Designs

## ASJC Scopus subject areas

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics