Enclosings of decompositions of complete multigraphs in 2‐factorizations

Abstract

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.

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

“…Feghali and Johnson [10] gave an example of a decomposition of 5K 4 that satisfies conditions (C1)-(C4) with r = 2 but that cannot be enclosed in some Hamiltonian decomposition of 6K 6 . However, we think that if n sufficiently large and m ≥ 2n − 2, then (C1)-(C4) are likely to always be sufficient conditions.…”

“…We require the following two lemmas. The first lemma is a generalization of [10,Proposition 2.3]. Its proof is similar to that of [10,Proposition 2.3] but is considerably shorter.…”

“…An investigation of Problem 1 in the case where λ < µ and t = r = 2 was recently instigated by Feghali and Johnson [10]. They obtained necessary and sufficient conditions for enclosing a given decomposition of λK n in some Hamiltonian decomposition of µK m in several cases:…”

Embedding connected factorizations

“…‚ Last but not least, enclosing decompositions [4,19,30,31], and highly edge-connected factorizations of complete graphs [27,25] have been studied. In the absence of any corresponding results for hypergraphs, we pose the following problems.…”