{
  "abstract": "A graph is said to be 1-universally rigid if, for any generic realization of its vertices on the real line, the realization cannot be deformed out of the line in any higher-dimensional Euclidean space while preserving all edge lengths. In this paper, we investigate the sparsity of 1-universally rigid graphs in different manners. First, we show that an Erdős-Rényi random graph $G(n,p)$ is 1-universally rigid with high probability whenever $p \\geq C\\log n \\cdot \\log\\log n/n$ for some sufficiently large constant $C>0$. These techniques also imply the existence of 1-universally rigid graphs of arbitrarily large girth, which resolves a problem suggested by Dantas e Moura, Jordán, and Silverman in a non-constructive manner. We also study how sparse a 1-universally rigid graph can be, proving that there exists an absolute constant $\\varepsilon>0$ such that every sufficiently large 1-universally rigid graph on $n$ vertices has at least $(3/2+\\varepsilon)n$ edges. This breaks the natural $3/2$ barrier arising from degree-2 extensions and makes progress towards a question of Jordán and Nguyen.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.41581",
  "author_total": 3,
  "authors": [
    {
      "name": "Ole Gabsdil"
    },
    {
      "name": "Ezra Hirschi"
    },
    {
      "name": "Julien Portier"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/cn2rtgx3p9m3v.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.41581",
  "fulltext_url": null,
  "id": "cn2rtgx3p9m3v",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.41581",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.41581",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/cn2rtgx3p9m3v.ris",
  "sources": [
    {
      "first_seen": "2026-11-02T12:21:20Z",
      "native_id": "2610.41581",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.41581"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "On sparse universally rigid graphs in $\\mathbb{R}$",
  "title_text": "On sparse universally rigid graphs in ℝ",
  "type": "preprint",
  "updated_at": "2026-11-04T15:20:52Z",
  "url": "https://research.draconitic.com/paper/cn2rtgx3p9m3v",
  "venue": "arXiv",
  "versions": []
}