Graph designs, packings and coverings of λKv with a graph of six vertices and containing a triangle

Let λKv be the complete multigraph with v vertices, where any two distinct vertices x and y are joined by λ edges (x, y). Let G be a finite simple graph. A G-design (G-packing, G-covering) of λKv, denoted by (v,G, λ)-GD ((v,G, λ)-PD, (v,G, λ)-CD), is a pair (X,B) where X is the vertex set of Kv and B is a collection of subgraphs of Kv, called blocks, such that each block is isomorphic to G and any two distinct vertices in Kv are joined in exactly (at most, at least) λ blocks of B. In this paper, we determine the existence spectrum for the G-designs of λKv, λ > 1, and construct the maximum packings and the minimum coverings of λKv with G for any positive integer λ, where the graph G has six vertices and contains a triangle.