{
  "id": "1611.03221",
  "version": "v1",
  "published": "2016-11-10T08:54:40.000Z",
  "updated": "2016-11-10T08:54:40.000Z",
  "title": "Indecomposable $1$-factorizations of the complete multigraph $λ K_{2n}$ for every $λ\\leq 2n$",
  "authors": [
    "Simona Bonvicini",
    "Gloria Rinaldi"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "A $1$-factorization of the complete multigraph $\\lambda K_{2n}$ is said to be indecomposable if it cannot be represented as the union of $1$-factorizations of $\\lambda_0 K_{2n}$ and $(\\lambda-\\lambda_0) K_{2n}$, where $\\lambda_0<\\lambda$. It is said to be simple if no $1$-factor is repeated. For every $n\\geq 9$ and for every $(n-2)/3\\leq\\lambda\\leq 2n$, we construct an indecomposable $1$-factorization of $\\lambda K_{2n}$ which is not simple. These $1$-factorizations provide simple and indecomposable $1$-factorizations of $\\lambda K_{2s}$ for every $s\\geq 18$ and $2\\leq\\lambda\\leq 2\\lfloor s/2\\rfloor-1$. We also give a generalization of a result by Colbourn et al. which provides a simple and indecomposable $1$-factorization of $\\lambda K_{2n}$, where $2n=p^m+1$, $\\lambda=(p^m-1)/2$, $p$ prime.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-11-10T08:54:40.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C70"
    ],
    "keywords": [
      "complete multigraph",
      "factorization",
      "indecomposable"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}