On nesting of G-decompositions of lambdaKv where G has four nonisolated vertices or les

The complete multigraph λKυ is said to have a G-decomposition if it is the union of edge disjoint subgraphs of Kυ each of them isomorphic to a fixed graph G. The spectrum problem for G-decompositions of λKυ that have a nesting was first considered in the case G = K3 by Colbourn and Colbourn (Ars Combin. 16 (1983) 27-34) and Stinson (Graphs and Combin. 1 (1985) 189 -191). For λ = 1 and G = Cm (the cycle of length m) this problem was studied in many papers, see Lindner and Rodger (in: J.H. Dinitz, D.R. Stinson (Eds.), Contemporary Design Theory: A Collection of Surveys, Wiley, New York, 1992, p. 325-369), Lindner et al. (Discrete Math. 77 (1989) 191-203), Lindner and Stinson (J. Combin. Math. Combin. Comput. 8 (1990) 147-157) for more details and references. For λ = 1 and G = Pk (the path of length k - 1) the analogous problem was considered in Milici and Quattrocchi (J. Combin. Math. Combin. Comput. 32 (2000) 115-127). In this paper we solve the spectrum problem of nested G-decompositions of λKυ for all the graphs G having four nonisolated vertices or less, leaving eight possible exceptions.