{
  "id": "2103.12255",
  "version": "v1",
  "published": "2021-03-23T02:02:42.000Z",
  "updated": "2021-03-23T02:02:42.000Z",
  "title": "On the number of $k$-gons in finite projective planes",
  "authors": [
    "Vladislav Taranchuk"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "Let $\\Pi$ be a projective plane of order $n$ and $\\Gamma_{\\Pi}$ be its Levi graph (the point-line incidence graph). For fixed $k \\geq 3$, let $c_{2k}(\\Gamma_{\\Pi})$ denote the number of $2k$-cycles in $\\Gamma_{\\Pi}$. In this paper we show that $$c_{2k}(\\Gamma_{\\Pi}) = \\frac{1}{2k}n^{2k} + O(n^{2k-2}), \\hspace{0.5cm} n \\rightarrow \\infty. $$ We also state a conjecture regarding the third and fourth largest terms in the asymptotic of the number of $2k$-cycles in $\\Gamma_{\\Pi}$. Let $\\text{ex}(v, C_{2k}, \\mathcal{C}_{\\text{odd}}\\cup \\{C_4\\})$ denote the greatest number of $2k$-cycles amongst all bipartite graphs of order $v$ and girth at least 6. As a corollary of the result above, we obtain $$ \\text{ex}(v, C_{2k}, \\mathcal{C}_{\\text{odd}}\\cup \\{C_4\\}) = \\left(\\frac{1}{2^{k+1}k}-o(1)\\right)v^k, \\hspace{0.5cm} v \\rightarrow \\infty.$$",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-03-23T02:02:42.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05B25",
      "05C35"
    ],
    "keywords": [
      "finite projective planes",
      "fourth largest terms",
      "point-line incidence graph",
      "levi graph",
      "greatest number"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}