{
  "abstract": "Alon and Mubayi asked whether, for every integer $k\\ge1$ and every $\\varepsilon>0$, there exists $r_0=r_0(k,\\varepsilon)$ such that every $r$-regular graph on $n$ vertices with $r\\ge r_0$ contains a $k$-regular subgraph covering at least $(1-\\varepsilon)n$ vertices. Previously, the conjecture was known for $k\\in\\{1,2\\}$ and for $k$ and $r$ both even. We answer this question affirmatively for general $k$ and $r$ with $\\varepsilon=O_k(r^{-1/2})$. For $k=2$ and every odd $r\\ge3$, we show that $\\varepsilon=1/(r^2-3)$ suffices which is best possible.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.19777",
  "author_total": 1,
  "authors": [
    {
      "name": "Varun Sivashankar"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/32qjesqkgjvfk.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.19777",
  "fulltext_url": null,
  "id": "32qjesqkgjvfk",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.19777",
  "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|Limits and Structures in Graph Theory"
    ]
  ],
  "openalex": "W7213589959",
  "pdf_url": "https://arxiv.org/pdf/2609.19777",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-17",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/32qjesqkgjvfk.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:23Z",
      "native_id": "2609.19777",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.19777"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Nearly Spanning Regular Subgraphs",
  "title_text": "Nearly Spanning Regular Subgraphs",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:23Z",
  "url": "https://research.draconitic.com/paper/32qjesqkgjvfk",
  "venue": "arXiv",
  "versions": []
}