{
  "abstract": "A set $S$ of vertices of a graph $G$ is a mutual-visibility set if every pair of vertices of $S$ is joined by a shortest path whose internal vertices avoid $S$, and the mutual-visibility chromatic number $χ_μ(G)$ is the least number of mutual-visibility sets into which $V(G)$ can be partitioned. Klavžar, Kuziak, Valenzuela Tripodoro and Yero (2025) proved that $χ_μ(G)\\leχ(G)$ whenever $G$ has diameter two, observed that equality holds for complete bipartite graphs, and asked whether there are any other diameter-two graphs attaining equality. We answer this question in the affirmative. We construct a graph $G_{11}$ of order 11 and diameter two with $χ_μ(G_{11})=χ(G_{11})=3$, and an infinite family of such graphs obtained from $G_{11}$ by replacing one vertex with pairwise non-adjacent copies of it.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.39430",
  "author_total": 1,
  "authors": [
    {
      "name": "Siddhesh Chaubal"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/79fgf7t5mzp4q.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.39430",
  "fulltext_url": null,
  "id": "79fgf7t5mzp4q",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.39430",
  "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.DM"
    ],
    [
      "arxiv",
      "math.CO"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.39430",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/79fgf7t5mzp4q.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:32Z",
      "native_id": "2610.39430",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.39430"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Diameter-two graphs whose mutual-visibility chromatic number equals their chromatic number",
  "title_text": "Diameter-two graphs whose mutual-visibility chromatic number equals their chromatic number",
  "type": "preprint",
  "updated_at": "2026-11-01T12:20:50Z",
  "url": "https://research.draconitic.com/paper/79fgf7t5mzp4q",
  "venue": "arXiv",
  "versions": []
}