{
  "abstract": "A unit-distance graph is a graph which admits a realisation in Euclidean space in which every edge has unit length. Imposing further geometric conditions on the non-edges gives a family of natural subclasses. Requiring that no two vertices lie at distance less than one gives penny and marble graphs: the contact graphs of collections of equal radii $d$-dimensional spheres with non-overlapping interiors for $d=2,3$. Requiring instead that the straight-line drawing in the plane be non-crossing gives matchstick graphs. Since any motion of a realisation must preserve these extra conditions, the rigidity and flexibility properties of the resulting frameworks differ from those of classical bar-joint rigidity theory. In this note we analyse various rigidity problems for penny and marble graphs, matchstick graphs and unit-distance graphs. In particular we answer a recent open problem on penny and marble graph rigidity and establish a link between penny graphs and the concept of NAC-colourings.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.06246",
  "author_total": 5,
  "authors": [
    {
      "name": "Sean Dewar"
    },
    {
      "name": "Georg Grasegger"
    },
    {
      "name": "Alison La Porta"
    },
    {
      "name": "Jan Legerský"
    },
    {
      "name": "Anthony Nixon"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dbgkx5dashamx.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.06246",
  "fulltext_url": null,
  "id": "dbgkx5dashamx",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.06246",
  "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.MG"
    ],
    [
      "openalex",
      "1703|Advanced Graph Theory Research"
    ]
  ],
  "openalex": "W7220694450",
  "pdf_url": "https://arxiv.org/pdf/2610.06246",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dbgkx5dashamx.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:10Z",
      "native_id": "2610.06246",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.06246"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.MG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "On rigidity properties of unit-distance graphs",
  "title_text": "On rigidity properties of unit-distance graphs",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:10Z",
  "url": "https://research.draconitic.com/paper/dbgkx5dashamx",
  "venue": "arXiv",
  "versions": []
}