{
  "abstract": "Maxwell observed that the graph of any rigid generic framework in $\\mathbb{R}^d$ on $n$ vertices has at least $dn-\\binom{d+1}{2}$ edges. In this article we prove that graphs whose complement has maximum degree at most two and no component isomorphic to a triangle or a square are rigid in the maximum dimension allowed by this observation. In particular, this determines the precise maximum dimension in which the graph obtained from a complete graph $K_{2m}$ by deleting a perfect matching is rigid, resolving a recent conjecture of Lew. We also deduce bounds on the rigidity of complements of bounded-degree graphs more generally, which significantly improve existing degree-based bounds.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.05058",
  "author_total": 3,
  "authors": [
    {
      "name": "John Haslegrave"
    },
    {
      "name": "Peleg Michaeli"
    },
    {
      "name": "Anthony Nixon"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0xqx34awg1f2e.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.05058",
  "fulltext_url": null,
  "id": "0xqx34awg1f2e",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.05058",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "1703|Advanced Graph Theory Research"
    ]
  ],
  "openalex": "W7211920303",
  "pdf_url": "https://arxiv.org/pdf/2609.05058",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-04",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0xqx34awg1f2e.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:18Z",
      "native_id": "2609.05058",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.05058"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Rigidity of complements of bounded-degree graphs",
  "title_text": "Rigidity of complements of bounded-degree graphs",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:18Z",
  "url": "https://research.draconitic.com/paper/0xqx34awg1f2e",
  "venue": "arXiv",
  "versions": []
}