{
  "id": "2409.03073",
  "version": "v1",
  "published": "2024-09-04T20:55:28.000Z",
  "updated": "2024-09-04T20:55:28.000Z",
  "title": "On the existence of Hamiltonian cycles in hypercubes",
  "authors": [
    "Gabriele Di Pietro",
    "Marco Ripà"
  ],
  "comment": "6 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "For each pair of positive integers $(a,b)$ such that $a \\geq 0$ and $b > 1$, the present paper provides a necessary and sufficient condition for the existence of Hamiltonian cycles visiting all the vertices of any $k$-dimensional grid $\\{0,1\\}^k \\subset \\mathbb{R}^k$ and whose associated Euclidean distance is equal to $\\sqrt{a^2+b^2}$. Our solution extends previously stated results in fairy chess on the existence of closed Euclidean $(a,b)$-leapers tours for $2 \\times 2 \\times \\cdots \\times 2$ chessboards, where the (Euclidean) knight identifies the $(1,2)$-leaper.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-09-04T20:55:28.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C12",
      "05C45",
      "05C38"
    ],
    "keywords": [
      "hamiltonian cycles",
      "hypercubes",
      "sufficient condition",
      "dimensional grid",
      "associated euclidean distance"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 6,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}