{
  "abstract": "By Tutte's classic theorem of 1956 that every 4-connected planar graph is Hamiltonian, every planar graph of order at least three with toughness greater than $\\frac{3}{2}$ is Hamiltonian. In 1999, Owens constructed a sequence of maximal planar graphs whose toughness approaches $\\frac{3}{2}$ from below and which do not contain even a 2-factor, and he asked whether there exists a maximal planar graph with toughness exactly $\\frac{3}{2}$ and with no 2-factor. In 2025, Shan constructed a $\\frac{3}{2}$-tough plane triangulation with no 2-factor. In that construction, there are many pairs of vertices of degree $3$ that have a common neighbor. By imposing a distance condition on the vertices of degree $3$, Hao, Ma, Shan, and Yang recently proved that every $\\frac{3}{2}$-tough plane triangulation of order at least three whose vertices of degree $3$ are pairwise at distance at least $3$ has a 2-factor, and they asked whether every such graph is Hamiltonian. We answer this question in the negative, and in fact prove the following stronger statement: for every positive integer $\\ell$, there exists a $\\frac{3}{2}$-tough non-Hamiltonian plane triangulation whose vertices of degree $3$ are pairwise at distance at least $\\ell$. Thus, although the distance condition guarantees the existence of a 2-factor, it does not guarantee that the graph is Hamiltonian: the essential obstruction to a Hamiltonian cycle is a certain local configuration involving a vertex of degree $3$, rather than the proximity of such configurations in the graph.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.05414",
  "author_total": 1,
  "authors": [
    {
      "name": "Songling Shan"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/d7keattye4h3r.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.05414",
  "fulltext_url": null,
  "id": "d7keattye4h3r",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.05414",
  "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": "W7211894398",
  "pdf_url": "https://arxiv.org/pdf/2609.05414",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-04",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/d7keattye4h3r.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:18Z",
      "native_id": "2609.05414",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.05414"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Non-Hamiltonian $\\frac{3}{2}$-Tough Plane Triangulations",
  "title_text": "Non-Hamiltonian 3/2-Tough Plane Triangulations",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:18Z",
  "url": "https://research.draconitic.com/paper/d7keattye4h3r",
  "venue": "arXiv",
  "versions": []
}