{
  "id": "2105.03304",
  "version": "v1",
  "published": "2021-05-07T14:53:09.000Z",
  "updated": "2021-05-07T14:53:09.000Z",
  "title": "Improved bounds for zeros of the chromatic polynomial on bounded degree graphs",
  "authors": [
    "Maurizio Moreschi",
    "Viresh Patel",
    "Guus Regts",
    "Ayla Stam"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "We prove that for any graph $G$ of maximum degree at most $\\Delta$, the zeros of its chromatic polynomial $\\chi_G(z)$ (in $\\mathbb{C}$) lie outside the disk of radius $5.02 \\Delta$ centered at $0$. This improves on the previously best known bound of approximately $6.91\\Delta$. In the case of graphs of high girth we can improve this. We prove that for every $g$ there is a constant $K_g$ such that for any graph $G$ of maximum degree at most $\\Delta$ and girth at least $g$, the zeros of its chromatic polynomial $\\chi_G(z)$ lie outside the disk of radius $K_g \\Delta$ centered at $0$ where $K_g \\to 1 + e \\approx 3.72$ as $g \\to \\infty$. Finally, we give improved bounds on the Fisher zeros of the partition function of the Ising model.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-05-07T14:53:09.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "82B20",
      "68W25"
    ],
    "keywords": [
      "chromatic polynomial",
      "bounded degree graphs",
      "maximum degree",
      "lie outside",
      "partition function"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}