{
  "id": "1810.12340",
  "version": "v1",
  "published": "2018-10-29T18:38:41.000Z",
  "updated": "2018-10-29T18:38:41.000Z",
  "title": "Enclosings of Decompositions of Complete Multigraphs in $2$-Edge-Connected $r$-Factorizations",
  "authors": [
    "John Asplund",
    "Pierre Charbit",
    "Carl Feghali"
  ],
  "comment": "17 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "A decomposition of a multigraph $G$ is a partition of its edges into subgraphs $G(1), \\ldots , G(k)$. It is called an $r$-factorization if every $G(i)$ is $k$-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 $1 \\leq i \\leq k$, $G(i)$ is a subgraph of $H(i)$. Feghali and Johnson gave necessary and sufficient conditions for a given decomposition of $\\lambda K_n$ to be enclosed in some $2$-edge-connected $r$-factorization of $\\mu K_{m}$ for some range of values for the parameters $n$, $m$, $\\lambda$, $\\mu$, $r$: $r=2$, $\\mu>\\lambda$ and either $m \\geq 2n-1$, or $m=2n-2$ and $\\mu = 2$ and $\\lambda=1$, or $n=3$ and $m=4$. In this paper, we generalize their result to every $r \\geq 2$ and $m \\geq 2n - 2$. We also give some sufficient conditions for enclosing a given decomposition of $\\lambda K_n$ in some $2$-edge-connected $r$-factorization of $\\mu K_{m}$ for every $r \\geq 3$ and $m = (2 - C)n$, where $C$ is a constant that depends only on $r$, $\\lambda$ and~$\\mu$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-10-29T18:38:41.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05B99"
    ],
    "keywords": [
      "decomposition",
      "complete multigraphs",
      "factorization",
      "sufficient conditions",
      "johnson gave necessary"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}