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 [G-decomposition of Kn, where G has four vertices or less, Discrete Math. 19 (1977) 113-120] proved that a kite-design of order n exists if and only if n≡0or1(mod8). 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≡0or1(mod8), 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.