On nesting of G-decompositions of λKv where G has four nonisolated vertices or less Original Research Article

The complete multigraph λKv is said to have a G-decomposition if it is the union of edge disjoint subgraphs of Kv each of them isomorphic to a fixed graph G. The spectrum problem for G-decompositions of λKv 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 λKv for all the graphs G having four nonisolated vertices or less, leaving eight possible exceptions.