More on decompositions of edge-colored complete graphs

Let GG be a family of graphs whose edges are colored with elements from a set RR of rr colors. We assume no two vertices of GG are joined by more than one edge of color ii for any i∈Ri∈R, for each G∈GG∈G. Kn(r) will denote the complete graph with rr edges joining any pair of distinct vertices, one of each of the rr colors. We describe necessary and asymptotically sufficient conditions on nn for the existence of a family DD of subgraphs of Kn(r), each of which is an isomorphic copy of some graph in GG, so that each edge of Kn(r) appears in exactly one of the subgraphs in DD.