Small embedding of an S-3(2, 4, u) into an S-lambda,(2, 4, u + omega)

Let H be a subgraph of a graph G. An H-design (U, C) of order u and index mu, is embedded into a G-design (V, B) of order v and index lambda if mu. <= lambda, U subset of V and there is an injective mapping.f : C -> B such that B is a subgraph of f (B) for every B epsilon C. The mapping f is called the embedding of (U, C) into (V, B). We determine, for every admissible value of u and lambda, the minimum value of w (except 12 values of (u, lambda)) such that every S-3(2, 4, u) can be embedded into an S-lambda(2, 4, u + w). This result implies that we determine also the minimum value of w such that there exists an S-lambda(2, 4, u + w) which embeds an E-2(u, 1), where E-2 is the graph with two parallel edges and without isolated vertices. (C) 2011 Elsevier B.V. All rights reserved.