{
  "abstract": "We resolve two open problems regarding uniquely colorable Cayley graphs posed by Klotz and Sander (2017). First, we construct an infinite family of uniquely 3-colorable integral circulant graphs with a clique number of 2. This provides a negative answer to Problem 3.6, which asks whether every uniquely colorable circulant graph satisfies $χ(G) = ω(G)$. Because verifying unique colorability inherently relies on the exact independence number, we demonstrate that traditional spectral bounds fail to tightly capture this parameter, necessitating a rigorous combinatorial proof based on exact structural isomorphisms. Second, we establish a general algebraic construction proving the existence of uniquely colorable Cayley graphs over nonabelian groups whose color classes are left cosets of strictly distinct subgroups. By utilizing right-coset partitions of non-normal subgroups, this result provides a definitive affirmative answer to Problem 2.4.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.03184",
  "author_total": 1,
  "authors": [
    {
      "name": "Milan Bašić"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0r9e66h4jwxx5.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.03184",
  "fulltext_url": null,
  "id": "0r9e66h4jwxx5",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.03184",
  "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",
      "math.CO"
    ],
    [
      "openalex",
      "2607|Finite Group Theory Research"
    ]
  ],
  "openalex": "W7208766586",
  "pdf_url": "https://arxiv.org/pdf/2609.03184",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0r9e66h4jwxx5.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:36Z",
      "native_id": "2609.03184",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.03184"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "On uniquely colorable Cayley graphs",
  "title_text": "On uniquely colorable Cayley graphs",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:36Z",
  "url": "https://research.draconitic.com/paper/0r9e66h4jwxx5",
  "venue": "arXiv",
  "versions": []
}