{
  "abstract": "Delcourt and Postle recently introduced the `discharging in the dual' proof technique, leading to a breakthrough proof of Nash-Williams' conjecture on triangle decompositions. Independently, and almost simultaneously, the first and second authors built on the `condensation' method of Bryant, Dukes, Horsley, Maenhaut and Montgomery to determine the decomposition threshold of all odd cycles of length at least $73$. In this note, we explain how the `discharging in the dual' and `condensation' methods can be combined to determine the decomposition threshold of cycles of any odd length, as well as the approximate decomposition threshold of a range of other $3$-chromatic graphs.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.41266",
  "author_total": 3,
  "authors": [
    {
      "name": "Bertille Granet"
    },
    {
      "name": "Daniel Horsley"
    },
    {
      "name": "Jeffrey N. Santrio"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4s8tvs4wsrn0d.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.41266",
  "fulltext_url": null,
  "id": "4s8tvs4wsrn0d",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.41266",
  "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",
      "math.CO"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.41266",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4s8tvs4wsrn0d.ris",
  "sources": [
    {
      "first_seen": "2026-11-02T12:21:20Z",
      "native_id": "2610.41266",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.41266"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "A note on decompositions into 3-chromatic graphs",
  "title_text": "A note on decompositions into 3-chromatic graphs",
  "type": "preprint",
  "updated_at": "2026-11-04T15:20:52Z",
  "url": "https://research.draconitic.com/paper/4s8tvs4wsrn0d",
  "venue": "arXiv",
  "versions": []
}