{
  "id": "2108.12669",
  "version": "v1",
  "published": "2021-08-28T16:05:59.000Z",
  "updated": "2021-08-28T16:05:59.000Z",
  "title": "Triangle-free planar graphs with at most $64^{n^{0.731}}$ 3-colorings",
  "authors": [
    "Zdeněk Dvořák",
    "Luke Postle"
  ],
  "comment": "4 pages, 1 figure",
  "categories": [
    "math.CO"
  ],
  "abstract": "Thomassen conjectured that triangle-free planar graphs have exponentially many 3-colorings. Recently, he disproved his conjecture by providing examples of such graphs with $n$ vertices and at most $2^{15n/\\log_2 n}$ 3-colorings. We improve his construction, giving examples of such graphs with at most $64^{n^{log_{9/2} 3}}<64^{n^{0.731}}$ 3-colorings. We conjecture this exponent is optimal.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-08-28T16:05:59.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C15"
    ],
    "keywords": [
      "triangle-free planar graphs",
      "conjecture",
      "construction"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 4,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}