{
  "id": "1610.04581",
  "version": "v1",
  "published": "2016-10-14T18:56:27.000Z",
  "updated": "2016-10-14T18:56:27.000Z",
  "title": "Nowhere-zero $3$-flow and $\\mathbb{Z}_3$-connectedness in Graphs with Four Edge-disjoint Spanning Trees",
  "authors": [
    "Miaomiao Han",
    "Hong-Jian Lai",
    "Jiaao Li"
  ],
  "comment": "14 pages, 3 figures",
  "categories": [
    "math.CO"
  ],
  "abstract": "Given a zero-sum function $\\beta : V(G) \\rightarrow \\mathbb{Z}_3$ with $\\sum_{v\\in V(G)}\\beta(v)=0$, an orientation $D$ of $G$ with $d^+_D(v)-d^-_D(v)= \\beta(v)$ in $\\mathbb{Z}_3$ for every vertex $v\\in V(G)$ is called a $\\beta$-orientation. A graph $G$ is $\\mathbb{Z}_3$-connected if $G$ admits a $\\beta$- orientation for every zero-sum function $\\beta$. Jaeger et al. conjectured that every $5$-edge-connected graph is $\\mathbb{Z}_3$-connected. A graph is $\\langle\\mathbb{Z}_3\\rangle$-extendable at vertex $v$ if any pre-orientation at $v$ can be extended to a $\\beta$-orientation of $G$ for any zero-sum function $\\beta$. We observe that if every $5$-edge-connected essentially $6$-edge-connected graph is $\\langle\\mathbb{Z}_3\\rangle$-extendable at any degree five vertex, then the above mentioned conjecture by Jaeger et al. holds as well. Furthermore, applying the partial flow extension method of Thomassen and of Lov\\'{a}sz et al., we prove that every graph with at least 4 edge-disjoint spanning trees is $\\mathbb{Z}_3$-connected. Consequently, every $5$-edge-connected essentially $23$-edge-connected graph is $\\langle\\mathbb{Z}_3\\rangle$-extendable at degree five vertex.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-10-14T18:56:27.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C21"
    ],
    "keywords": [
      "edge-disjoint spanning trees",
      "zero-sum function",
      "edge-connected graph",
      "nowhere-zero",
      "connectedness"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}