Small embedding of an S3(2,4,u)S3(2,4,u) into an Sλ(2,4,u+w)Sλ(2,4,u+w)

Let HH be a subgraph of a graph GG. An HH-design (U,C)(U,C) of order uu and index μμ is embedded into a GG-design (V,B)(V,B) of order vv and index λλ if μ≤λμ≤λ, U⊆VU⊆V and there is an injective mapping f:C→Bf:C→B such that BB is a subgraph of f(B)f(B) for every B∈CB∈C. The mapping ff is called the embedding of (U,C)(U,C) into (V,B)(V,B). We determine, for every admissible value of uu and λλ, the minimum value of ww (except 1212 values of (u,λ)(u,λ)) such that every S3(2,4,u)S3(2,4,u) can be embedded into an Sλ(2,4,u+w)Sλ(2,4,u+w). This result implies that we determine also the minimum value of w such that there exists an Sλ(2,4,u+w)Sλ(2,4,u+w) which embeds an E2(u,1)E2(u,1), where E2E2 is the graph with two parallel edges and without isolated vertices.