Minimum embedding of a -design into a balanced incomplete block design of index

Let H be a subgraph of G . An H -design ( V , C ) of order v and index μ is embedded into a G -design ( X , B ) of order v + w and index λ if μ ≤ λ , V ⊆ X and there is an injective mapping f : C → B such that B is subgraph of f ( B ) for every B ∈ C . ery pair of positive integers v , λ , (except when λ = 3 and v = 30 , 34 , 42 , 46 , 54 , 58 , 66 or λ = 5 and v = 19 ) we determine the minimum value of w such that there exists a balanced incomplete block design of order v + w , index λ and block-size 4 which embeds a P 4 -design of order v and index μ = 1 ( P 4 denotes the path of length 3).