{
  "abstract": "A connected graph is called $\\ell_1$-embeddable if its shortest-path metric admits an isometric embedding into the $\\ell_1$-space; equivalently, in the finite case, it admits a scale embedding into a hypercube. This paper gives a complete parameter-level classification of $\\ell_1$-embeddability for regular quadrilateral maps on the torus and the Klein bottle. We treat the two toroidal families $Q_{k,m,r}$ and $Q_{k,m,e}$, obtained from suitable identifications of finite square grids, and, using the known isomorphisms among Klein-bottle quadrilateral maps, the three essential Klein-bottle families $Q_{k,m,a}$, $Q_{k,m,b}$, and $Q_{k,m,h}$. The $\\ell_1$-embeddable toroidal maps are exactly $Q_{k,m,0}, Q_{k,2,1}, Q_{k,1,e}, Q_{5,2,e}, Q_{2i,2,e} (i\\ge 2)$, and the $\\ell_1$-embeddable Klein-bottle maps are exactly $ Q_{2,2,a}, Q_{4,2,a}, Q_{2,m,b}, Q_{1,2,h}.$ All other members of these five families are not $\\ell_1$-embeddable.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.03770",
  "author_total": 3,
  "authors": [
    {
      "name": "Guangfu Wang"
    },
    {
      "name": "Wenchao Cong"
    },
    {
      "name": "Chunxiao Xu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8seg1t3ezmdvw.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.03770",
  "fulltext_url": null,
  "id": "8seg1t3ezmdvw",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.03770",
  "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": "W7220665327",
  "pdf_url": "https://arxiv.org/pdf/2610.03770",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8seg1t3ezmdvw.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:10Z",
      "native_id": "2610.03770",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.03770"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "$\\ell_{1}$-Embeddability of Regular Quadrilateral Maps on the Torus and the Klein Bottle",
  "title_text": "ℓ_1-Embeddability of Regular Quadrilateral Maps on the Torus and the Klein Bottle",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:10Z",
  "url": "https://research.draconitic.com/paper/8seg1t3ezmdvw",
  "venue": "arXiv",
  "versions": []
}