{
  "abstract": "We prove Sabidussi's compatibility conjecture. Let $G$ be a finite connected multigraph in which every vertex has even degree and the minimum degree is at least four, and let $T$ be an Euler tour of $G$. The edges of $G$ can be partitioned into circuits (connected $2$-regular subgraphs) so that no circuit contains two edges used consecutively anywhere in $T$. In fact, the edges can be four-coloured so that every such pair receives different colours and every colour class has even degree at every vertex. We use a counting argument based on the Chevalley-Warning theorem to show that a four-colouring with the required properties exists. Splitting each colour class into circuits then gives the desired compatible decomposition. Formalization in Lean 4 is also available in the author's github.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.13225",
  "author_total": 1,
  "authors": [
    {
      "name": "Nikolay Ulyanov"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/c1ybedykcsn7m.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.13225",
  "fulltext_url": null,
  "id": "c1ybedykcsn7m",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.13225",
  "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": "W7168575575",
  "pdf_url": "https://arxiv.org/pdf/2607.13225",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/c1ybedykcsn7m.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:10Z",
      "native_id": "2607.13225",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.13225"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Graph Puzzles III.1: A Proof of Sabidussi's Compatibility Conjecture",
  "title_text": "Graph Puzzles III.1: A Proof of Sabidussi's Compatibility Conjecture",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:10Z",
  "url": "https://research.draconitic.com/paper/c1ybedykcsn7m",
  "venue": "arXiv",
  "versions": []
}