Halving transversal designs

A factor H of a transversal design TD(k,n) = (V,𝒢, ℬ︁), where V is the set of points, 𝒢 the set of groups of size n and ℬ︁ the set of blocks of size k, is a triple (V,𝒢, 𝒟) such that 𝒟 is a subset of ℬ︁. A halving of a TD (k, n) is a pair of factors H_i = (V, 𝒢, 𝒟_i), i = 1,2 such that 𝒟₁ ∪ 𝒟₂ = ℬ︁, 𝒟₁ ∩ 𝒟₂ = ∅︁ and H₁ is isomorphic to H₂. A path of length q is a sequence x₀, x₁,…,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