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

AbstractThe 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

α-Resolvable λ-fold G-designs

A λ-fold G-design is said to be α-resolvable if its blocks can be partitioned into classes such that every class contains each vertex exactly α times. In this paper we study the existence problem of an α-resolvable λ-fold G-design oforder v in the case when G is any connected subgraph of K_4 and prove that the necessary conditions for its existence are also sufficient

On uniformly resolvable (C4,K1,3)(C_4,K_{1,3})-designs

In this paper we consider the uniformly resolvable decompositions of the complete graph $K_v$ minus a 1-factor $(K_v − I)$ into subgraphs where each resolution class contains only blocks isomorphic to the same graph. We completely determine the spectrum for the case in which all the resolution classes consist of either 4-cycles or 3-stars

Uniformly resolvable decompositions of Kv in 1-factors and 4-stars

If X is a connected graph, then an X-factor of a larger graph is a spanning subgraph in which all of its components are isomorphic to X. A uniformly resolvable {X, Y }-decomposition of the complete graph Kv is an edge decomposition of Kv into exactly r X-factors and s Y -factors. In this article we determine necessary and sufficient conditions for when the complete graph Kv has a uniformly resolvable decompositions into 1-factors and K1,4-factors