The spectrum of Meta(K4 - e > K3 + e, λ) with any λ1

Abstract

Let (X, Β) be a (λK_v, G₁)-design and G₂ a subgraph of G₁. Define sets Β(G₂) and D(G₂\G₂) as follows: for each block B ε Β, partition B into copies of G₂and G₂ \ G ₂ and place the copy of G₂ in Β(G₂) and the edges belonging to the copy of G₂\ G₂ in D(G₂ \ G ₂). If the edges belonging to D(G₁\ G₂) can be assembled into a collection D(G₂) of copies of G₂, then (X, Β(G₂) ∪ D(G₂)) is a (λK_v, G₂)-design, called a metamorphosis of the (λK_v, G₁)design (X, Β). For brevity we denote such (λK _v, G₁)-design (X, Β) with a metamorphosis of (λK_v,G₂)-design (λK_v, (G ₂) ∪ D(G₂)) by (λK_v, G₁ > G₂)-design. Let Meta(G₁ > G₂, λ) denote the set of all integers v such that there exists a (λK _v, G₁ > G₂)-design. In this paper we completely determine the set M eta(K₄ - e > K₃ + e, λ) for any λ.