{
  "abstract": "For every $d\\ge4$, every prescribed connectivity $r\\ge d$, and every $\\varepsilon>0$, we construct a simplicial $d$-polytope whose graph has connectivity exactly $r$, is nontraceable, and has toughness less than $\\varepsilon$. Thus the classical theorems of Whitney (1931) and Tutte (1956), which guarantee hamiltonicity under sufficiently high connectivity in dimension three, have no higher-dimensional counterpart. This answers a question posed by Bojan Mohar in 2001 on whether sufficiently high connectivity guarantees hamiltonicity of $4$-polytopal graphs and the connectivity questions raised by Ikegami, Maezawa, and Zamfirescu in 2021. For fixed $d\\ge4$, $r\\ge d$, and integer $k\\ge2$, the examples can be chosen to have no closed spanning walk with at most $k$ visits per vertex, no spanning tree with degrees at most $k$, and nontraceable prisms. For fixed $d$ and $r$, we obtain graphs on $n$ vertices whose longest paths and cycles contain $O(\\sqrt n)$ vertices. Consequently, the family of simplicial $d$-polytopes with connectivity exactly $r$ has Grünbaum--Walther shortness coefficient zero and shortness exponent at most $1/2$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.12836",
  "author_total": 1,
  "authors": [
    {
      "name": "Guillermo Pineda-Villavicencio"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/cmfxfff2jsqa1.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.12836",
  "fulltext_url": null,
  "id": "cmfxfff2jsqa1",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.12836",
  "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": "W7221507263",
  "pdf_url": "https://arxiv.org/pdf/2610.12836",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-09",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/cmfxfff2jsqa1.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:28:40Z",
      "native_id": "2610.12836",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.12836"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Connectivity, toughness, and hamiltonicity of simplicial polytopes",
  "title_text": "Connectivity, toughness, and hamiltonicity of simplicial polytopes",
  "type": "preprint",
  "updated_at": "2026-10-30T12:28:40Z",
  "url": "https://research.draconitic.com/paper/cmfxfff2jsqa1",
  "venue": "arXiv",
  "versions": []
}