{
  "abstract": "We prove that $Z(P(n,3))=8$ for every $n\\ge13$ and $Z(P(n,4))=10$ for every $n\\ge18$, with both stabilization thresholds sharp, and give the exact values for all smaller admissible $n$. The first equality proves Krishnan's Conjecture 5; his Proposition 4 supplies its upper bound. The lower bounds use the external boundary of the first forcing sources: ten vertices have boundary at least eight in $P(n,3)$ for $n\\ge14$, and sixteen vertices have boundary at least ten in $P(n,4)$ for $n\\ge20$. Deletion within an empty column interval reduces these assertions to finite checks, and forts settle the remaining cases. The $k=3$ stabilization theorem has a Lean 4 formalization; the $k=4$ proof uses 89 boundary formulas and 11 fort formulas with separately checked DRAT certificates, together with explicit forcing sequences.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.24059",
  "author_total": 3,
  "authors": [
    {
      "name": "Arnav Krishnan"
    },
    {
      "name": "Haobo Ma"
    },
    {
      "name": "Wenlin Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9anwmxxvbmtd8.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.24059",
  "fulltext_url": null,
  "id": "9anwmxxvbmtd8",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.24059",
  "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",
      "1703|Advanced Graph Theory Research"
    ]
  ],
  "openalex": "W7223045868",
  "pdf_url": "https://arxiv.org/pdf/2610.24059",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-16",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9anwmxxvbmtd8.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:57Z",
      "native_id": "2610.24059",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.24059"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Sharp zero forcing numbers for the generalized Petersen graphs $P(n,3)$ and $P(n,4)$",
  "title_text": "Sharp zero forcing numbers for the generalized Petersen graphs P(n,3) and P(n,4)",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:44Z",
  "url": "https://research.draconitic.com/paper/9anwmxxvbmtd8",
  "venue": "arXiv",
  "versions": []
}