## Abstract

A factor H of a transversal design TD(k, n) = (V, script G sign, ℬ), where V is the set of points, script G sign the set of groups of size n and ℬ the set of blocks of size k, is a triple (V, script G sign, script D) such that script D is a subset of ℬ. A halving of a TD(k, n) is a pair of factors H_{i} = (V, script G sign, script D_{i}), i = 1, 2 such that script D_{i} ∪ script D_{2} = ℬ, script D_{1} ∩ script D_{2} = Ø and H_{1} is isomorphic to H_{2}. A path of length q is a sequence x_{0},x_{1}, . . . ,x_{q} of points such that for each i = 1,2, . . . , q the points x_{i-1} and x_{i} belong to a block B_{i} and no point appears more than once. The distance between points x and y in a factor H is the length of the shortest path from x to y. The diameter of a connected factor H is the maximum of the set of distances among all pairs of points of H. We prove that a TD(3, n) is halvable into isomorphic factors of diameter d only if d = 2, 3, 4, or ∞ and we completely determine for which values of n there exists such a halvable TD(3, n). We also show that if any group divisible design with block size at least 3 is decomposed into two factors with the same finite diameter d, then d ≤ 4.

Original language | English (US) |
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Pages (from-to) | 83-99 |

Number of pages | 17 |

Journal | Journal of Combinatorial Designs |

Volume | 8 |

Issue number | 2 |

DOIs | |

State | Published - 2000 |

## Keywords

- Complete multipartite graphs
- Diameter
- Isomorphic factors
- Self complementary factors
- Transversal designs