{
  "abstract": "A backbite on a Hamiltonian path of a graph $G$ adds an edge of $G$ from an endpoint $u$ of the path to a vertex $v$ and deletes an edge of the path at $v$ so that the result is again a Hamiltonian path. We study the connectivity of the reconfiguration graph $R(G)$, whose vertices are the Hamiltonian paths of $G$, with two adjacent when a single backbite carries one to the other. We prove that $R(G)$ is connected whenever $G$ contains the square of a Hamiltonian path as a spanning subgraph, and that any two Hamiltonian paths can be reconfigured using $O(n^2)$ backbites. Combining the theorem with known results on spanning squared paths gives connectivity for every graph with $δ(G)\\ge\\lceil\\frac{2n-1}3\\rceil$, for the circulants $\\mathrm{Circ}(n;S)$ with $a,2a\\in S$ for some $a$ coprime to $n$, and, with high probability, for the random graph $G(n,p)$ with $p\\ge c/\\sqrt n$, for a sufficiently large constant $c$. In the other direction, graphs with arbitrarily high connectivity may have disconnected reconfiguration graphs. For every $n\\ge8$ and every $1\\le m\\le\\lfloor n/2\\rfloor-3$ we construct a graph on $n$ vertices with $κ=λ=δ=m$ whose reconfiguration graph is disconnected, with at least $(n-2m-4)!$ components. We conjecture that $δ(G)\\ge n/2$ suffices for connectivity.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.05068",
  "author_total": 1,
  "authors": [
    {
      "name": "Albi Kazazi"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5d15kppxpcpt3.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.05068",
  "fulltext_url": null,
  "id": "5d15kppxpcpt3",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.05068",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "1703|Advanced Graph Theory Research"
    ]
  ],
  "openalex": "W7220701550",
  "pdf_url": "https://arxiv.org/pdf/2610.05068",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-04",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5d15kppxpcpt3.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:10Z",
      "native_id": "2610.05068",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.05068"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Connectivity of the reconfiguration graph of Hamiltonian paths",
  "title_text": "Connectivity of the reconfiguration graph of Hamiltonian paths",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:10Z",
  "url": "https://research.draconitic.com/paper/5d15kppxpcpt3",
  "venue": "arXiv",
  "versions": []
}