{
  "abstract": "For $n$ points in the plane and $\\varepsilon>0$, we construct a $(1+\\varepsilon)$-spanner with $O(n/\\varepsilon)$ edges in which every edge has $\\tilde{O}(1/\\varepsilon^3)$ crossings, the total number of crossings is $\\tilde{O}(n/\\varepsilon^4)$, and the ratio between the lengths of any two crossing edges is $O(1/\\varepsilon^2)$. Our spanner construction substantially improves on the previous upper bound for the number of crossings in a $(1+\\varepsilon)$-spanner, and it is the first spanner construction that ensures $O(1)$ crossings per edge for any constant $\\varepsilon>0$. In contrast, we construct $n$ points in the plane for which every $(1+\\varepsilon)$-spanner has $Ω(n/\\varepsilon^3)$ crossings, $n$ points for which every $(1+\\varepsilon)$-spanner has an edge with $Ω(1/\\varepsilon^{5/2})$ crossings, and 4 points for which every $(1+\\varepsilon)$-spanner contains two crossing edges where one is $Ω(1/\\varepsilon)$ times longer than the other.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.25040",
  "author_total": 2,
  "authors": [
    {
      "name": "Kelvin Luu"
    },
    {
      "name": "Csaba D. Tóth"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/a6mqn9mygjrn7.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.25040",
  "fulltext_url": null,
  "id": "a6mqn9mygjrn7",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.25040",
  "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",
      "cs.CG"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "1704|Computational Geometry and Mesh Generation"
    ]
  ],
  "openalex": "W7171629450",
  "pdf_url": "https://arxiv.org/pdf/2607.25040",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-17",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/a6mqn9mygjrn7.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:57Z",
      "native_id": "2607.25040",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.25040"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Geometric $(1+\\varepsilon)$-Spanners with Few Crossings",
  "title_text": "Geometric (1+ε)-Spanners with Few Crossings",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:44Z",
  "url": "https://research.draconitic.com/paper/a6mqn9mygjrn7",
  "venue": "arXiv",
  "versions": []
}