Embedding handcuffed designs in D-designs,

Let D be the triangle with attached edge (i. e. D is the graph having vertices {a0, a1, a2, a3} and edges {a0, a1}, {a0, a2}, {a1, a2}, {a0, a3}). J.C. Bermond and J. Schönheim [1], proved that a D-design of order n exists if and only if n ≡ 0 or 1 (mod 8). Let (W, C) be a nontrivial D-design of order n, n ≥ 8, and let V ⊂ W , |V | = v < n. We say that a handcuffed design (V,P) of order v and block size s ([3]) is embedded in (W, C) if there is an injective mapping f : P → C such that B is a subgraph of f (B) for every B ∈ P. For each n ≡ 0 or 1 (mod 8), we determine the spectrum of all the integers v such that there is a nontrivial handcuffed design of order v and block size s embedded in a D-design of order n. AMS classification: 05B05.