{
  "id": "2205.08971",
  "version": "v1",
  "published": "2022-05-18T14:52:50.000Z",
  "updated": "2022-05-18T14:52:50.000Z",
  "title": "Powers of Hamilton cycles in dense graphs perturbed by a random geometric graph",
  "authors": [
    "Alberto Espuny Díaz",
    "Joseph Hyde"
  ],
  "comment": "arXiv admin note: text overlap with arXiv:2102.02321",
  "categories": [
    "math.CO"
  ],
  "abstract": "Let $G$ be a graph obtained as the union of some $n$-vertex graph $H_n$ with minimum degree $\\delta(H_n)\\geq\\alpha n$ and a $d$-dimensional random geometric graph $G^d(n,r)$. We investigate under which conditions for $r$ the graph $G$ will a.a.s. contain the $k$-th power of a Hamilton cycle, for any choice of $H_n$. We provide asymptotically optimal conditions for $r$ for all values of $\\alpha$, $d$ and $k$. This has applications in the containment of other spanning structures, such as $F$-factors.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-05-18T14:52:50.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "hamilton cycle",
      "dense graphs",
      "dimensional random geometric graph",
      "minimum degree",
      "th power"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}