{
  "id": "1608.06961",
  "version": "v1",
  "published": "2016-08-24T20:29:32.000Z",
  "updated": "2016-08-24T20:29:32.000Z",
  "title": "Enclosings of Decompositions of Complete Multigraphs in 2-Factorizations",
  "authors": [
    "Carl Feghali",
    "Matthew Johnson"
  ],
  "comment": "19 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "Let $k$, $\\lambda$ and $\\mu$ be positive integers. A decomposition of a multigraph $ \\lambda G$ into edge-disjoint subgraphs $G_1, \\ldots , G_k$ is said to be \\emph{enclosed} by a decomposition of a multigraph $\\mu H$ into edge-disjoint subgraphs $H_1, \\ldots , H_k$ if $\\mu > \\lambda$ and $G_i$ is a subgraph of $H_i$, $1 \\leq i \\leq k$. In this paper we initiate the study of when a decomposition can be enclosed by a decomposition that consists of spanning subgraphs. A decomposition of a graph is a 2-factorization if each subgraph is 2-regular and is Hamiltonian if each subgraph is a Hamiltonian cycle. Let $n$ and $m$ be positive integers. We give necessary and sufficient conditions for enclosing a decomposition of $\\lambda K_n$ in a $2$-factorization of $\\mu K_{n+m}$ whenever $\\mu>\\lambda$ and $m \\geq n-2$. We also give necessary and sufficient conditions for enclosing a decomposition of $\\lambda K_n$ in a Hamiltonian decomposition of $\\mu K_{n+m}$ whenever $\\mu > \\lambda$ and $m \\geq n-1$, or $\\mu > \\lambda$, $n=3$ and $m=1$, or $\\mu = 2$, $\\lambda=1$ and $m=n-2$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-08-24T20:29:32.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C70"
    ],
    "keywords": [
      "complete multigraphs",
      "edge-disjoint subgraphs",
      "sufficient conditions",
      "positive integers",
      "hamiltonian decomposition"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}