Halving transversal designs

A factor H of a transversal design TD ( k,n ) = ( V ,, ℬ), where V is the set of points, the set of groups of size n and ℬ the set of blocks of size k , is a triple ( V ,, ) such that is a subset of ℬ. A halving of a TD ( k, n ) is a pair of factors H i = ( V , , i ), i = 1,2 such that 1 ∪ 2 = ℬ, 1 ∩ 2 = ∅ 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. © 2000 John Wiley & Sons, Inc. J Combin Designs 8: 83–99, 2000