Halving transversal designs

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 D₂ = ℬ, script D₁ ∩ script D₂ = Ø 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.