{
  "id": "1812.09833",
  "version": "v1",
  "published": "2018-12-24T05:17:06.000Z",
  "updated": "2018-12-24T05:17:06.000Z",
  "title": "Circular Flows in Planar Graphs",
  "authors": [
    "Daniel W. Cranston",
    "Jiaao Li"
  ],
  "comment": "18 pages, 6 figures, plus 2.5 page appendix with 2 figures, comments welcome",
  "categories": [
    "math.CO"
  ],
  "abstract": "For integers $a\\ge 2b>0$, a \\emph{circular $a/b$-flow} is a flow that takes values from $\\{\\pm b, \\pm(b+1), \\dots, \\pm(a-b)\\}$. The Planar Circular Flow Conjecture states that every $2k$-edge-connected planar graph admits a circular $(2+\\frac{2}{k})$-flow. The cases $k=1$ and $k=2$ are equivalent to the Four Color Theorem and Gr\\\"otzsch's 3-Color Theorem. For $k\\ge 3$, the conjecture remains open. Here we make progress when $k=4$ and $k=6$. We prove that (i) {\\em every 10-edge-connected planar graph admits a circular 5/2-flow} and (ii) {\\em every 16-edge-connected planar graph admits a circular 7/3-flow.} The dual version of statement (i) on circular coloring was previously proved by Dvo\\v{r}\\'ak and Postle (Combinatorica 2017), but our proof has the advantages of being much shorter and avoiding the use of computers for case-checking. Further, it has new implications for antisymmetric flows. Statement (ii) is especially interesting because the counterexamples to Jaeger's original Circular Flow Conjecture are 12-edge-connected nonplanar graphs that admit no circular 7/3-flow. Thus, the planarity hypothesis of (ii) is essential.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-12-24T05:17:06.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C15"
    ],
    "keywords": [
      "planar graph admits",
      "planar circular flow conjecture states",
      "jaegers original circular flow conjecture",
      "conjecture remains open"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}