{
  "id": "1711.04232",
  "version": "v1",
  "published": "2017-11-12T03:57:44.000Z",
  "updated": "2017-11-12T03:57:44.000Z",
  "title": "Pairs of disjoint cycles",
  "authors": [
    "Serguei Norine",
    "Robin Thomas",
    "Hein van der Holst"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "Let $G=(V,E)$ be a finite undirected graph. Let $\\mathbb{Z}\\langle V\\rangle$ denote free $\\mathbb{Z}$-module generated by the vertices of $G$. Let $\\mathbb{Z}\\langle E\\rangle$ denote the free $\\mathbb{Z}$-module generated by the oriented edges of $G$. A $2$-cycle of $G$ is a bilinear form $d : \\mathbb{Z}\\langle E\\rangle\\times \\mathbb{Z}\\langle E\\rangle\\to \\mathbb{Z}$ such for each edge $e$ of $G$, $d(e,\\cdot)$ and $d(\\cdot,e)$ are circulations, and $d(e,f) = 0$ whenever $e$ and $f$ have a common vertex. The $2$-cycles of a graph $G$ are in one-to-one correspondence with the homology classes in the second homology group of the deleted product of $G$. We show that each $2$-cycle is a linear combination of three special types of $2$-cycles: cycle-pair $2$-cycles, Kuratowski $2$-cycles, and quad $2$-cycles. Furthermore, we show that each skew-symmetric $2$-cycles is a linear combination of two special types of $2$-cycles: skew-symmetric cycle-pair $2$-cycles and skew-symmetric quad $2$-cycles.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-11-12T03:57:44.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "disjoint cycles",
      "special types",
      "linear combination",
      "second homology group",
      "denote free"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}