{
  "abstract": "For every positive integer $k$, we prove that the homogeneous cut semigroup of every $K_5$-minor-free graph is saturated at height $k$ if and only if every planar $k$-graph is $k$-edge-colourable. Here a $k$-graph is a loopless $k$-regular multigraph in which every odd vertex cut has size at least $k$. A triangle expansion of a cubic plane dual converts cut decompositions into perfect-matching decompositions. The converse uses symmetric difference with a fixed perfect matching. The equivalence identifies the normality conjecture for $K_5$-minor-free cut polytopes with Seymour's planar edge-colouring conjecture. In particular, the known cases $k\\leq 8$ give saturation through height eight.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.12732",
  "author_total": 1,
  "authors": [
    {
      "name": "SeungJu Lee"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0bytv2j8apehr.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.12732",
  "fulltext_url": null,
  "id": "0bytv2j8apehr",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.12732",
  "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": "W7212701026",
  "pdf_url": "https://arxiv.org/pdf/2609.12732",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-11",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0bytv2j8apehr.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:11Z",
      "native_id": "2609.12732",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.12732"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Degreewise Cut-Semigroup Saturation for $K_5$-Minor-Free Graphs and Seymour's Planar Edge-Colouring Conjecture",
  "title_text": "Degreewise Cut-Semigroup Saturation for K_5-Minor-Free Graphs and Seymour's Planar Edge-Colouring Conjecture",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:11Z",
  "url": "https://research.draconitic.com/paper/0bytv2j8apehr",
  "venue": "arXiv",
  "versions": []
}