{
  "abstract": "Albertson's conjecture states that every graph of chromatic number r has crossing number at least that of the complete graph K_r. We prove the conjecture for all r. The proof combines new clique-sensitive edge bounds for critical graphs, maximum-clique sampling, and a reserved immersion construction. For graphs of order close to r, we construct edge-disjoint routes while preserving a complete bipartite subgraph whose forced crossings compensate for those introduced when the immersion is smoothed into a drawing. For larger orders, we derive explicit edge bounds using independent-triple deletion and structural properties of critical graphs, then combine them with sampling inequalities that account for crossings within a maximum clique. The remaining numerical inequalities are certified using exact rational arithmetic, with a reproducible certificate and standalone Python verifier.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.01682",
  "author_total": 1,
  "authors": [
    {
      "name": "Ankan Sadhu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5erf02pfjahr1.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.01682",
  "fulltext_url": null,
  "id": "5erf02pfjahr1",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.01682",
  "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",
      "cs.CG"
    ],
    [
      "arxiv",
      "cs.DM"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "1703|Advanced Graph Theory Research"
    ]
  ],
  "openalex": "W7207683961",
  "pdf_url": "https://arxiv.org/pdf/2609.01682",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-12",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5erf02pfjahr1.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:15Z",
      "native_id": "2609.01682",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.01682"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "A Proof of Albertson's Conjecture",
  "title_text": "A Proof of Albertson's Conjecture",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:15Z",
  "url": "https://research.draconitic.com/paper/5erf02pfjahr1",
  "venue": "arXiv",
  "versions": []
}