Let k, m, n, λ, and μ be positive integers. A decomposition of λKn into edge‐disjoint subgraphs G1,…,Gk is said to be enclosed by a decomposition of μKm into edge‐disjoint subgraphs H1,…,Hk if μ>λ and, after a suitable labeling of the vertices in both graphs, λKn is a subgraph of μKm and Gi is a subgraph of Hi for all i=1,⋯,k. In this paper, we continue the study of enclosings of given decompositions by decompositions that consist of spanning subgraphs. A decomposition of a graph is a 2‐factorization if each subgraph is 2‐regular and spanning, and is Hamiltonian if each subgraph is a Hamiltonian cycle. We give necessary and sufficient conditions for the existence of a 2‐factorization of μKn+m that encloses a given decomposition of λKn whenever μ>λ and m≥n−2. We also give necessary and sufficient conditions for the existence of a Hamiltonian decomposition of μKn+m that encloses a given decomposition of λKn whenever μ>λ and either m≥n−1 or n=3 and m=1, or μ=2, λ=1, and m=n−2.
Journal of Combinatorial Designs – Wiley
Published: Jan 1, 2018
Keywords: ; ;
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