Decomposition of λKv into some graph with six vertices and seven edges

Let λKv be the complete multigraph with v vertices. Let G be a finite simple graph. A G-design of λKv, denoted by (v,G,λ)-GD, 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 λ blocks of B. In this paper, the existence of (v,Gi,λ)-GD is discussed, for λ>1 and 1⩽i⩽16, where the Gi are specified graphs with six vertices and seven edges. The existence is completely established for all except the graph G16.