{
  "id": "2301.07467",
  "version": "v1",
  "published": "2023-01-18T12:21:24.000Z",
  "updated": "2023-01-18T12:21:24.000Z",
  "title": "Many Hamiltonian subsets in large graphs with given density",
  "authors": [
    "Stijn Cambie",
    "Jun Gao",
    "Hong Liu"
  ],
  "comment": "11 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "A set of vertices in a graph is a Hamiltonian subset if it induces a subgraph containing a Hamiltonian cycle. Kim, Liu, Sharifzadeh and Staden proved that among all graphs with minimum degree $d$, $K_{d+1}$ minimises the number of Hamiltonian subsets. We prove a near optimal lower bound that takes also the order and the structure of a graph into account. For many natural graph classes, it provides a much better bound than the extremal one ($\\approx 2^{d+1}$). Among others, our bound implies that an $n$-vertex $C_4$-free graphs with minimum degree $d$ contains at least $n2^{d^{2-o(1)}}$ Hamiltonian subsets.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-01-18T12:21:24.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C35",
      "05C38",
      "05C48"
    ],
    "keywords": [
      "hamiltonian subset",
      "large graphs",
      "minimum degree",
      "natural graph classes",
      "optimal lower bound"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}