{
  "abstract": "Baste, Furst, Henning, Mohr, and Rautenbach conjectured that every finite regular graph of positive degree satisfies \\(γ(G) \\leq γ_e(G)\\), where \\(γ\\) is the domination number and \\(γ_e\\) is the edge domination number, equivalently the minimum cardinality of a maximal matching. We show that the conjecture is false already for cubic graphs. The counterexample is a previously public 50-vertex cubic graph that had been used to refute the stronger independent-domination inequality \\(i(G) \\leq γ_e(G)\\). For this graph we prove \\(γ(G) = 16 > 15 = γ_e(G)\\). The equality \\(γ_e(G) = 15\\) has a short counting proof, and a dominating set of order 16 is displayed explicitly. For the lower bound \\(γ(G) \\geq 16\\), we give a self-contained exact reduction: after fixing which of the 20 clause vertices lie in a putative dominating set, the remaining problem is a finite set-cover problem on the 30 literal vertices. We enumerate all \\(2^{20} = 1,048,576\\) clause subsets, derive two explicit lower bounds, and solve exactly the 5,931 residual cases by a recurrence stated in the paper. The complete case counts and minima are displayed, and a short standard-library Python implementation is included in an appendix. A separate 893,049-node proof-tree certificate and a direct graph search provide independent verification. Thus the regular-graph conjecture is disproved. Combined with Gupta's recent theorem that every cubic graph on at most 48 vertices satisfies the conjectured inequality, the example is order-minimal among cubic counterexamples.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.10783",
  "author_total": 1,
  "authors": [
    {
      "name": "Koyar Afrasyab"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/b8qw1057zz069.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.10783",
  "fulltext_url": null,
  "id": "b8qw1057zz069",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.10783",
  "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",
      "1703|Advanced Graph Theory Research"
    ]
  ],
  "openalex": "W7212308336",
  "pdf_url": "https://arxiv.org/pdf/2609.10783",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-09",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/b8qw1057zz069.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:30Z",
      "native_id": "2609.10783",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.10783"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "A 50-Vertex Cubic Counterexample to the Domination-versus-Edge-Domination Conjecture",
  "title_text": "A 50-Vertex Cubic Counterexample to the Domination-versus-Edge-Domination Conjecture",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:30Z",
  "url": "https://research.draconitic.com/paper/b8qw1057zz069",
  "venue": "arXiv",
  "versions": []
}