On Nesting of -Decompositions of Λ Where Has Four Nonisolated Vertices or Less

The complete multigraph λ is said to have a -decomposition if it is the union of edge disjoint subgraphs of each of them isomorphic to a fixed graph . The spectrum problem for -decompositions of λ that have a nesting was first considered in the case = by C.J.Colbourn and M.J.Colbourn [4] and D.R. Stinson [15]. For λ = 1 and = (the cycle of length ) this problem was studied in many papers, see [9, 10, 11] for more details and references. For λ = 1 and = (the path of length —1) the analogous problem was considered in [13].In this paper we solve the spectrum problem of nested -decompositions of λ for all the graphs having four nonisolated vertices or less, leaving eight possible exceptions