{
  "id": "2405.20766",
  "version": "v1",
  "published": "2024-05-31T12:45:11.000Z",
  "updated": "2024-05-31T12:45:11.000Z",
  "title": "Long cycles and spectral radii in planar graphs",
  "authors": [
    "Ping Xu",
    "Huiqiu Lin",
    "Longfei Fang"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "There is a rich history of studying the existence of cycles in planar graphs. The famous Tutte theorem on the Hamilton cycle states that every 4-connected planar graph contains a Hamilton cycle. Later on, Thomassen (1983), Thomas and Yu (1994) and Sanders (1996) respectively proved that every 4-connected planar graph contains a cycle of length $n-1, n-2$ and $n-3$. Chen, Fan and Yu (2004) further conjectured that every 4-connected planar graph contains a cycle of length $\\ell$ for $\\ell\\in\\{n,n-1,\\ldots,n-25\\}$ and they verified that $\\ell\\in \\{n-4, n-5, n-6\\}$. When we remove the ``4-connected\" condition, how to guarantee the existence of a long cycle in a planar graph? A natural question asks by adding a spectral radius condition: What is the smallest constant $C$ such that for sufficiently large $n$, every graph $G$ of order $n$ with spectral radius greater than $C$ contains a long cycle in a planar graph? In this paper, we give a stronger answer to the above question. Let $G$ be a planar graph with order $n\\geq 1.8\\times 10^{17}$ and $k\\leq \\lfloor\\log_2(n-3)\\rfloor-8$ be a non-negative integer, we show that if $\\rho(G)\\geq \\rho(K_2\\vee(P_{n-2k-4}\\cup 2P_{k+1}))$ then $G$ contains a cycle of length $\\ell$ for every $\\ell\\in \\{n-k, n-k-1, \\ldots, 3\\}$ unless $G\\cong K_2\\vee(P_{n-2k-4}\\cup 2P_{k+1})$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-05-31T12:45:11.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C50",
      "05C35",
      "05C45"
    ],
    "keywords": [
      "long cycle",
      "planar graph contains",
      "hamilton cycle states",
      "spectral radius condition",
      "spectral radius greater"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}