{
  "abstract": "The $r$-neighbor bootstrap percolation process on a graph $G$ is a vertex-activation process that begins with a set of initially active vertices. In each subsequent round, every inactive vertex having at least $r$ active neighbors becomes active. Denote by $m(G,r)$ the minimum number of initially active vertices whose activation eventually spreads to all vertices of $G$. Let \\(\\mathbbmsl{O}_k\\) and \\(\\mathbbmsl{B}_k\\) be the odd graph and the bipartite odd graph, respectively, both defined on a ground set of size \\(2k+1\\). In this article, among other results, we prove that $m(\\mathbbmsl{O}_k, 2)=\\lceil(k^2+3k+2)/4\\rceil$ and $m(\\mathbbmsl{B}_k, 2)=\\lceil(k^2+2k+3)/2\\rceil$. The former result confirms a conjecture posed in 2021 by Grippo, Pastine, Torres, Valencia-Pabon, and Vera.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2606.03687",
  "author_total": 3,
  "authors": [
    {
      "name": "Ali Mohammadian"
    },
    {
      "name": "Sina Rezaie Zareie"
    },
    {
      "name": "Behruz Tayfeh-Rezaie"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7b6fbf7rykmsv.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2606.03687",
  "fulltext_url": null,
  "id": "7b6fbf7rykmsv",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2606.03687",
  "language": null,
  "licence": "CC0-1.0",
  "licence_label": "CC0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/publicdomain/zero/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "2607|Limits and Structures in Graph Theory"
    ]
  ],
  "openalex": "W7163358272",
  "pdf_url": "https://arxiv.org/pdf/2606.03687",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-18",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7b6fbf7rykmsv.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:57Z",
      "native_id": "2606.03687",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2606.03687"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "$\\boldsymbol{2}$-Neighbor Bootstrap Percolation on Odd Graphs",
  "title_text": "2-Neighbor Bootstrap Percolation on Odd Graphs",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:44Z",
  "url": "https://research.draconitic.com/paper/7b6fbf7rykmsv",
  "venue": "arXiv",
  "versions": []
}