{
  "abstract": "A rotation system on $n$ elements assigns to each element a cyclic order of the other $n-1$ elements. A four-element subset is non-planar if its induced rotation system cannot be realized by a crossing-free drawing of $K_4$. As a combinatorial strengthening of Hill's conjecture on the crossing number of the complete graph, Archdeacon conjectured that every rotation system on $n$ elements has at least $H(n)=\\frac{1}{4} \\lfloor\\frac {n}{2}\\rfloor \\lfloor\\frac{n-1}{2}\\rfloor \\lfloor\\frac{n-2}{2}\\rfloor \\lfloor\\frac{n-3}{2}\\rfloor$ non-planar four-element subsets. We computationally verify Archdeacon's conjecture for $n\\leq 10$ and show that every extremal rotation system in these orders is realizable by a simple drawing. With computer assistance, we prove that every rotation system on $n$ elements has at least $(8/9 - o(1)) H(n)$ non-planar four-element subsets. We also present a proof by hand for a weaker lower bound of $(2/3-o(1)) H(n)$. Finally, extending recent work of Felsner on antipodal pairs in drawings, we show that Archdeacon's conjecture holds for antipodally shellable rotation systems.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.11599",
  "author_total": 2,
  "authors": [
    {
      "name": "Arahat Chikkatur"
    },
    {
      "name": "Ji Zeng"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6de45qk18vnvh.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.11599",
  "fulltext_url": null,
  "id": "6de45qk18vnvh",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.11599",
  "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.CG"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "1703|Advanced Graph Theory Research"
    ]
  ],
  "openalex": "W7212284971",
  "pdf_url": "https://arxiv.org/pdf/2609.11599",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-10",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6de45qk18vnvh.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:30Z",
      "native_id": "2609.11599",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.11599"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Some results on Archdeacon's conjecture for rotation systems",
  "title_text": "Some results on Archdeacon's conjecture for rotation systems",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:30Z",
  "url": "https://research.draconitic.com/paper/6de45qk18vnvh",
  "venue": "arXiv",
  "versions": []
}