{
  "abstract": "A graph $G$ is det-extremal if $|\\operatorname{det} A|=\\operatorname{per} A$ for its adjacency matrix $A$. Det-extremal cubic bipartite graphs arise in the study of Pólya's permanent problem, and McCuaig characterized the $3$-connected ones as vertex-sums of copies of the Heawood graph. The total domatic number of a graph is the largest number of pairwise disjoint total dominating sets. Characterization of the cubic graphs with total domatic number $1$ has been a long-standing open problem. In this paper, we prove that a connected cubic graph is det-extremal if and only if its total domatic number is $1$. We further show that McCuaig's characterization extends to all $3$-connected cubic graphs, and that every connected det-extremal cubic graph has girth $3$, $5$ or $6$. We also prove that a connected det-extremal cubic non-bipartite graph has at least $28$ vertices, and that this bound is best possible. Through this correspondence, these results carry over to cubic graphs with total domatic number $1$. In addition, in the language of configurations, our results imply that every triangle-free $3$-configuration has a blocking set.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.33902",
  "author_total": 3,
  "authors": [
    {
      "name": "Myungho Choi"
    },
    {
      "name": "Hyemin Kwon"
    },
    {
      "name": "Boram Park"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/084jzy2w24cm1.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.33902",
  "fulltext_url": null,
  "id": "084jzy2w24cm1",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.33902",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "1703|Advanced Graph Theory Research"
    ]
  ],
  "openalex": "W7214711568",
  "pdf_url": "https://arxiv.org/pdf/2609.33902",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-27",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/084jzy2w24cm1.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2609.33902",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.33902"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Det-extremal cubic graphs and the total domatic number",
  "title_text": "Det-extremal cubic graphs and the total domatic number",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:17Z",
  "url": "https://research.draconitic.com/paper/084jzy2w24cm1",
  "venue": "arXiv",
  "versions": []
}