Abstract: | A factor H of a transversal design TD(k,n) = (V,??, ??), where V is the set of points, ?? the set of groups of size n and ?? the set of blocks of size k, is a triple (V,??, ??) such that ?? is a subset of ??. A halving of a TD (k, n) is a pair of factors Hi = (V, ??, ??i), i = 1,2 such that ??1 ∪ ??2 = ??, ??1 ∩ ??2 = ?? and H1 is isomorphic to H2. A path of length q is a sequence x0, x1,…,xq of points such that for each i = 1, 2,…, q the points xi‐1 and xi belong to a block Bi 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 |