{
  "abstract": "Let $G$ be a graph together with a total order $\\prec$ on its edges. We say that $\\prec$ is realizable in $\\mathbb{R}^d$ if there is a placement of the vertices of $G$ in $\\mathbb{R}^d$ such that the Euclidean lengths of the edges induce exactly the order $\\prec$. Almendra-Hernández and Martínez-Sandoval proved that every total order on the edges of the complete graph $K_n$ is realizable in $\\mathbb{R}^{n-2}$. We show that the same is not true for the disjoint union of two complete graphs: for every $n\\geq 3$ there is a total order on the edges of $K_n\\sqcup K_n$ that is not realizable in $\\mathbb{R}^{n-2}$, but is in $\\mathbb{R}^{n-1}$. Surprisingly, the realizability of an order on a disconnected graph is not determined by its restrictions to the connected components. We also study realizability on the real line: we characterize which disjoint unions of two cycles are realizable, and estimate the largest number of edges an $n$-vertex graph can have while all of its edge-orders remain realizable on the line. In general dimension, we show that the largest number of edges of an $n$-vertex graph all of whose edge-orders are realizable in $\\mathbb{R}^d$ is $dn+O\\!\\left(dn/\\ln(dn)\\right)$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.15853",
  "author_total": 4,
  "authors": [
    {
      "name": "Gerardo L. Maldonado"
    },
    {
      "name": "Leonardo Martínez-Sandoval"
    },
    {
      "name": "Miguel Raggi"
    },
    {
      "name": "Edgardo Roldán-Pensado"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ewxv7ydwcanb5.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.15853",
  "fulltext_url": null,
  "id": "ewxv7ydwcanb5",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.15853",
  "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": "W7203668031",
  "pdf_url": "https://arxiv.org/pdf/2608.15853",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-19",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ewxv7ydwcanb5.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:27Z",
      "native_id": "2608.15853",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.15853"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Disconnected graphs and extremal bounds for realizable distance orders",
  "title_text": "Disconnected graphs and extremal bounds for realizable distance orders",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:27Z",
  "url": "https://research.draconitic.com/paper/ewxv7ydwcanb5",
  "venue": "arXiv",
  "versions": []
}