Embedding path designs into kite systems

Let D be the triangle with an attached edge (i. e. D is the “kite”, a graph having vertices {a0, a1, a2, a3} and edges {a0, a1}, {a0, a2}, {a1, a2}, {a0, a3}). Bermond and Schönheim [6] proved that a kite-design of order n exists if and only if n ≡ 0 or 1 (mod 8). Let (W, C) be a nontrivial kite-design of order n ≥ 8, and let V ⊂ W with |V | = v < n. A path design (V,P) of order v and block size s is embedded into (W, C) if there is an injective mapping f : P → C such that B is an induced subgraph of f (B) for every B ∈ P. For each n ≡ 0 or 1 (mod 8), we determine the spectrum of all integers v such that there is a nontrivial path design of order v and block size 3 embedded into a kite-design of order n. AMS classification: 05B05.