The spectrum of Meta(K3+e>P4,λ) and Meta(K3+e>H4,λ) with any λλ

Let (X,B)(X,B) be a (λKv,G1)(λKv,G1)-design and G2G2 a subgraph of G1G1. Define sets B(G2)B(G2) and D(G1∖G2)D(G1∖G2) as follows: for each block B∈BB∈B, partition BB into copies of G2G2 and G1∖G2G1∖G2 and place the copy of G2G2 in B(G2)B(G2) and the edges belonging to the copy of G1∖G2G1∖G2 in D(G1∖G2)D(G1∖G2). If the edges belonging to D(G1∖G2)D(G1∖G2) can be assembled into a collection D(G2)D(G2) of copies of G2G2, then (X,B(G2)∪D(G2))(X,B(G2)∪D(G2)) is a (λKv,G2)(λKv,G2)-design, called a metamorphosis   of the (λKv,G1)(λKv,G1)-design (X,B)(X,B). For brevity we denote such (λKv,G1)(λKv,G1)-design (X,B)(X,B) with a metamorphosis into (λKv,G2)(λKv,G2)-design (X,B(G2)∪D(G2))(X,B(G2)∪D(G2)) by (λKv,G1>G2)(λKv,G1>G2)-design. Let Meta(G1>G2,λ) denote the set of all integers vv such that there exists a (λKv,G1>G2)(λKv,G1>G2)-design. In this paper we completely determine the set Meta(K3+e>P4,λ) or Meta(K3+e>H4,λ) when the admissible conditions are satisfied, for any λλ.