4-Cycle decompositions of $$( \lambda +m)K_{v+u} {\setminus } \lambda K_v$$ ( λ + m ) K v + u \ λ K v

Newman, Nicholas

doi:10.1007/s10623-013-9904-6

Des. Codes Cryptogr.

2013

DOI: 10.1007/s10623-013-9904-6

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4-Cycle decompositions of $$( \lambda +m)K_{v+u} {\setminus } \lambda K_v$$ ( λ + m ) K v + u \ λ K v

Nicholas Newman

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Cited by 2 publications

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Enclosings of decompositions of complete multigraphs in 2‐factorizations

Feghali

Johnson

2018

J of Combinatorial Designs

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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.

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Enclosings of decompositions of complete multigraphs in 2‐factorizations

Feghali

Johnson

2018

J of Combinatorial Designs

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Enclosings of decompositions of complete multigraphs in 2-edge-connected r-factorizations

Asplund

Charbit

Feghali

2019

Discrete Mathematics

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A decomposition of a multigraph G is a partition of its edges into subgraphs G(1), . . . , G(k). It is called an r-factorization if every G(i) is r-regular and spanning.If G is a subgraph of H, a decomposition of G is said to be enclosed in a decomposition of H if, for every 1 ≤ i ≤ k, G(i) is a subgraph of H(i).Feghali and Johnson gave necessary and sufficient conditions for a given decomposition of λK n to be enclosed in some 2-edge-connected r-factorization of µK m for some range of values for the parameters n, m, λ, µ, r: r = 2, µ > λ and either m ≥ 2n − 1, or m = 2n − 2 and µ = 2 and λ = 1, or n = 3 and m = 4. We generalize their result to every r ≥ 2 and m ≥ 2n − 2. We also give some sufficient conditions for enclosing a given decomposition of λK n in some 2-edge-connected r-factorization of µK m for every r ≥ 3 and m = (2 − C)n, where C is a constant that depends only on r, λ and µ.

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4-Cycle decompositions of $$( \lambda +m)K_{v+u} {\setminus } \lambda K_v$$ ( λ + m ) K v + u \ λ K v

Cited by 2 publications

References 10 publications

Enclosings of decompositions of complete multigraphs in 2‐factorizations

Enclosings of decompositions of complete multigraphs in 2‐factorizations

Enclosings of decompositions of complete multigraphs in 2-edge-connected r-factorizations

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