{
  "id": "1809.02799",
  "version": "v1",
  "published": "2018-09-08T13:20:01.000Z",
  "updated": "2018-09-08T13:20:01.000Z",
  "title": "A note on the edge partition of graphs containing either a light edge or an alternating 2-cycle",
  "authors": [
    "Xin Zhang",
    "Bei Niu"
  ],
  "comment": "This is a very preliminary version! If you find any topes or mistakes, please fell free to let us now. This paper is used for communication, and will not be published as it is in a journal",
  "categories": [
    "math.CO",
    "cs.DM"
  ],
  "abstract": "Let $\\mathcal{G}_{\\alpha}$ be a hereditary graph class (i.e, every subgraph of $G_{\\alpha}\\in \\mathcal{G}_{\\alpha}$ belongs to $\\mathcal{G}_{\\alpha}$) such that every graph $G_{\\alpha}$ in $\\mathcal{G}_{\\alpha}$ has minimum degree at most 1, or contains either an edge $uv$ such that $d_{G_{\\alpha}}(u)+d_{G_{\\alpha}}(v)\\leq \\alpha$ or a 2-alternating cycle. It is proved that every graph in $\\mathcal{G}_{\\alpha}$ ($\\alpha\\geq 5$) with maximum degree $\\Delta$ can be edge-partitioned into two forests $F_1$, $F_2$ and a subgraph $H$ such that $\\Delta(F_i)\\leq \\max\\{2,\\lceil\\frac{\\Delta-\\alpha+6}{2}\\rceil\\}$ for $i=1,2$ and $\\Delta(H)\\leq \\alpha-5$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-09-08T13:20:01.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "edge partition",
      "light edge",
      "graphs containing",
      "hereditary graph class",
      "alternating"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}