The spectrum for a class of graph designs

Abstract

Let K _υ be the complete graph with v vertices. A G-design of K _υ, denoted by G-GD(υ), is a partition of K _υ into subgraphs so that each subgraph is isomorphic to G. In this paper, we will completely solve the case for G = K ₄∪K ₂ and prove that the necessary conditions υ ≡ 0,1 (mod 7) and υ ≥ 7 for the existence of G-GD(υ) are also sufficient except for υ ∈ {7,8}.