{
  "abstract": "Let $D(n)$ be the maximum number of unit-square diagonals in an $n\\times n$ square array such that no two selected diagonals share an endpoint and no unit square contains both diagonals. We prove that, for every odd positive integer $n$, $D(n)=n(n+1)/2+\\lfloor(n+1)/6\\rfloor$. The matching lower bound was established by Boyland, Roth, Pintér, Laukó, Schoenfield, and Wasielewski. We obtain the upper bound by transforming any admissible selection $M$ in a $(2k-1)\\times(2k-1)$ square array into an isometric path cover of the $k\\times k$ vertex grid with at most $2k^2-|M|$ paths. An auxiliary matching decomposes the original vertices into alternating paths. On each such path, all segments between blocks are either of type $da$ or of type $bc$. Projecting these paths to the block grid gives coordinate-monotone sequences, which become geodesics when diagonal steps are expanded. The Fisher--Fitzpatrick theorem then yields the sharp upper bound. We also describe a Lean verification of the upper-bound reduction, with the grid-cover theorem supplied as an explicit hypothesis.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.33042",
  "author_total": 1,
  "authors": [
    {
      "name": "Luyu Song"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/96tkjqedpevkv.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.33042",
  "fulltext_url": null,
  "id": "96tkjqedpevkv",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.33042",
  "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",
      "math.CO"
    ],
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7223948741",
  "pdf_url": "https://arxiv.org/pdf/2610.33042",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-16",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/96tkjqedpevkv.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:40Z",
      "native_id": "2610.33042",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.33042"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Nonintersecting Unit Diagonals in Odd Square Arrays",
  "title_text": "Nonintersecting Unit Diagonals in Odd Square Arrays",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:40Z",
  "url": "https://research.draconitic.com/paper/96tkjqedpevkv",
  "venue": "arXiv",
  "versions": []
}