{
  "abstract": "A single-track Gray code of length $n$ is a cyclic Gray code of $n$-bit words in which every column is a cyclic shift of the first. For prime $n$ the longest possible period is $2^n-2$. Whether it is attained for every prime has been open since Hiltgen, Paterson and Brandestini introduced these codes in 1996, and optimal codes were known for primes $n\\le 19$. We show that a conjecture of Degni and Drisko (2007) on the reflected Gray code order of necklaces implies optimal single-track Gray codes for every prime. Their list of necklaces, conjecturally a Gray path for prime $n$, closes into a Hamilton cycle of the necklace graph after $(n-3)/2$ explicit Pósa rotations, and such a cycle lifts to an optimal code. This implication is formalized in Lean 4. Together with computation it gives optimal single-track Gray codes for every odd prime $n\\le 47$: for $n\\le 37$ explicit codes checked word by word by a program independent of the generator, and for $n=41$, $43$ and $47$ codes obtained from the implication after the Degni-Drisko property was verified at these lengths by two independently written programs.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.24758",
  "author_total": 1,
  "authors": [
    {
      "name": "Chris Dzoba"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/e58z6829vmy35.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.24758",
  "fulltext_url": null,
  "id": "e58z6829vmy35",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.24758",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.DM"
    ],
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      "arxiv",
      "cs.IT"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.IT"
    ],
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7223003104",
  "pdf_url": "https://arxiv.org/pdf/2610.24758",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-17",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/e58z6829vmy35.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:57Z",
      "native_id": "2610.24758",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.24758"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Optimal single-track Gray codes from Gray-ordered necklaces",
  "title_text": "Optimal single-track Gray codes from Gray-ordered necklaces",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:44Z",
  "url": "https://research.draconitic.com/paper/e58z6829vmy35",
  "venue": "arXiv",
  "versions": []
}