{
  "abstract": "We propose and study a multi-labeling generalization of the $k$-radio labeling problem of graphs: given a graph $G$ and integers $k$ and $m$, assign to each vertex of $G$ a set of $m$ integer labels such that distinct labels assigned to a vertex differ by at least $k+1$, and the labels of any two distinct vertices $x$ and $y$ pairwise differ by at least $k+1-d(x,y)$, with $d(x,y)$ being the distance between vertices $x$ and $y$. We prove general bounds on the minimum span of such a labeling and more refined results for the case $k=1$. For paths and cycles, we derive sharp bounds on the minimum span when $k$ is small. We finally propose a conjecture for the minimum span on the path and for an almost-tight bound on the cycle.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.25863",
  "author_total": 1,
  "authors": [
    {
      "name": "Olivier Togni"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/b77zsg2nf8vj1.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.25863",
  "fulltext_url": null,
  "id": "b77zsg2nf8vj1",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.25863",
  "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",
      "cs.DM"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "1703|Graph Labeling and Dimension Problems"
    ]
  ],
  "openalex": "W7222996590",
  "pdf_url": "https://arxiv.org/pdf/2610.25863",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-19",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/b77zsg2nf8vj1.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:57Z",
      "native_id": "2610.25863",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.25863"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "On Multi-label radio Labelings of Graphs",
  "title_text": "On Multi-label radio Labelings of Graphs",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:44Z",
  "url": "https://research.draconitic.com/paper/b77zsg2nf8vj1",
  "venue": "arXiv",
  "versions": []
}