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 {a(0), a(1), a(2), a(3)} and edges {a(0), a(1)), {a(0), a(2)}, {a(1), a(2)), {a(0), a(3)}). Bermond and Schonheim [G-decomposition of K-n, 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 equivalent to 0 or 1 (mod 8). Let (W, C) be a nontrivial kite-design of order n >= 8, and let V subset of W with vertical bar V vertical bar = v < n. A path design (V, 9) 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 is an element of P. For each n &3bond; 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. (c) 2005 Elsevier B.V. All rights reserved.