{
  "abstract": "Inspect the vertices of a finite simple graph in uniformly random order, accepting each vertex if none of its neighbors has previously been accepted. Let $X_G$ be the number of accepted vertices. For every triangle-free graph with $n\\ge2$ vertices and $e$ edges, we prove $\\operatorname{Var}(X_G)\\le e((n-2)/n)^2$, with equality precisely for edgeless graphs and connected stars. In particular, among trees of a given order, the star uniquely maximizes the variance, with value $(n-1)(n-2)^2/n^2$. Known expected vertex-deletion stability already yields the elementary baseline $\\operatorname{Var}(X_G)\\le e$. We obtain the sharp finite-order refinement by combining a stronger centered first-choice estimate with a triangle-free edge-count identity in the law of total variance.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.14826",
  "author_total": 1,
  "authors": [
    {
      "name": "Mubin Shaikh"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8ayhrk3fx48cs.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.14826",
  "fulltext_url": null,
  "id": "8ayhrk3fx48cs",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.14826",
  "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"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2607|Limits and Structures in Graph Theory"
    ]
  ],
  "openalex": "W7213245294",
  "pdf_url": "https://arxiv.org/pdf/2609.14826",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-13",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8ayhrk3fx48cs.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:29Z",
      "native_id": "2609.14826",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.14826"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Variance of random greedy independent sets in triangle-free graphs",
  "title_text": "Variance of random greedy independent sets in triangle-free graphs",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:29Z",
  "url": "https://research.draconitic.com/paper/8ayhrk3fx48cs",
  "venue": "arXiv",
  "versions": []
}