{
  "abstract": "A graph G is uniquely C_t-saturated if G contains no cycle of length t and, for every edge e of the complement, G+e contains exactly one cycle of length t; it is nontrivial if it has at least t vertices. Wenger and West proved that no nontrivial uniquely C_6- or C_7-saturated graphs exist, and conjectured the same for every t >= 6; the case t=8 was verified but never published, and t >= 9 has remained open. We prove the conjecture for t=9: there is no nontrivial uniquely C_9-saturated graph. The proof is a case analysis on the length L of a longest even cycle of length at most 12 (L is 4, 6, 8, 10, or 12). The cases L=12, L=10, and L=4 are settled by hand; for L=6 and L=8, hand classifications of the components outside the cycle reduce each case to a bounded finite family of configurations, eliminated by a short, replayable computer enumeration with known-answer controls and completeness certified by zero cap hits. Along the way we prove, in sharpened form, the t=9 instance of a structural lemma Wenger and West stated without proof.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.33035",
  "author_total": 1,
  "authors": [
    {
      "name": "Fayzan Khan"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3pzgf5yz7agfb.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.33035",
  "fulltext_url": null,
  "id": "3pzgf5yz7agfb",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.33035",
  "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": "W7214758478",
  "pdf_url": "https://arxiv.org/pdf/2609.33035",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-27",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3pzgf5yz7agfb.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2609.33035",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.33035"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "There are no nontrivial uniquely C_9-saturated graphs",
  "title_text": "There are no nontrivial uniquely C_9-saturated graphs",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:17Z",
  "url": "https://research.draconitic.com/paper/3pzgf5yz7agfb",
  "venue": "arXiv",
  "versions": []
}