{
  "id": "1406.1554",
  "version": "v2",
  "published": "2014-06-06T01:38:06.000Z",
  "updated": "2014-07-18T06:10:45.000Z",
  "title": "A note on nowhere-zero 3-flow and Z_3-connectivity",
  "authors": [
    "Fuyuan Chen",
    "Bo Ning"
  ],
  "comment": "10 pages. Typos corrected",
  "categories": [
    "math.CO"
  ],
  "abstract": "There are many major open problems in integer flow theory, such as Tutte's 3-flow conjecture that every 4-edge-connected graph admits a nowhere-zero 3-flow, Jaeger et al.'s conjecture that every 5-edge-connected graph is $Z_3$-connected and Kochol's conjecture that every bridgeless graph with at most three 3-edge-cuts admits a nowhere-zero 3-flow (an equivalent version of 3-flow conjecture). Thomassen proved that every 8-edge-connected graph is $Z_3$-connected and therefore admits a nowhere-zero 3-flow. Furthermore, Lov$\\acute{a}$sz, Thomassen, Wu and Zhang improved Thomassen's result to 6-edge-connected graphs. In this paper, we prove that: (1) Every 4-edge-connected graph with at most seven 5-edge-cuts admits a nowhere-zero 3-flow. (2) Every bridgeless graph containing no 5-edge-cuts but at most three 3-edge-cuts admits a nowhere-zero 3-flow. (3) Every 5-edge-connected graph with at most five 5-edge-cuts is $Z_3$-connected. Our main theorems are partial results to Tutte's 3-flow conjecture, Kochol's conjecture and Jaeger et al.'s conjecture, respectively.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2014-07-18T06:10:45.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C21",
      "05C40"
    ],
    "keywords": [
      "nowhere-zero",
      "kochols conjecture",
      "major open problems",
      "integer flow theory",
      "bridgeless graph"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1406.1554C"
    }
  }
}