Nonexistences of the Packing and Covering Designs for the Join Graph of K1 and C4 with a Pendent Edge

Let λK_v be the complete multigraph with v vertices. A G-design G-GD_λ(v) of λK_v is a pair (X,B), where X is the vertex set of K_v and B is a collection of subgraphs of K_v, called blocks, such that each block is isomorphic to and any two distinct vertices in G are joined in K_v exact(at most, at least) l blocks of B. In this paper, we will discuss nonexistences and constructions of the maximum packing designs and the minimum covering designs for the join graph of K₁ and C₄ with a pendent edge for λ = 1.