{
  "id": "2101.05911",
  "version": "v1",
  "published": "2021-01-14T23:44:19.000Z",
  "updated": "2021-01-14T23:44:19.000Z",
  "title": "Counting paths, cycles and blow-ups in planar graphs",
  "authors": [
    "Christopher Cox",
    "Ryan R. Martin"
  ],
  "comment": "31 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "For a planar graph $H$, let $\\operatorname{\\mathbf{N}}_{\\mathcal P}(n,H)$ denote the maximum number of copies of $H$ in an $n$-vertex planar graph. In this paper, we prove that $\\operatorname{\\mathbf{N}}_{\\mathcal P}(n,P_7)\\sim{4\\over 27}n^4$, $\\operatorname{\\mathbf{N}}_{\\mathcal P}(n,C_6)\\sim(n/3)^3$, $\\operatorname{\\mathbf{N}}_{\\mathcal P}(n,C_8)\\sim(n/4)^4$ and $\\operatorname{\\mathbf{N}}_{\\mathcal P}(n,K_4\\{1\\})\\sim(n/6)^6$, where $K_4\\{1\\}$ is the $1$-subdivision of $K_4$. In addition, we obtain significantly improved upper bounds on $\\operatorname{\\mathbf{N}}_{\\mathcal P}(n,P_{2m+1})$ and $\\operatorname{\\mathbf{N}}_{\\mathcal P}(n,C_{2m})$ for $m\\geq 4$. For a wide class of graphs $H$, the key technique developed in this paper allows us to bound $\\operatorname{\\mathbf{N}}_{\\mathcal P}(n,H)$ in terms of an optimization problem over weighted graphs.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-01-14T23:44:19.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C35"
    ],
    "keywords": [
      "counting paths",
      "vertex planar graph",
      "optimization problem",
      "maximum number",
      "wide class"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 31,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}