{
  "abstract": "Albertson's conjecture asserts that every finite simple graph $G$ with $χ(G) \\ge r$ satisfies $\\operatorname{cr}(G) \\ge \\operatorname{cr}(K_r)$. Building on Cranston's verification for $r \\le 24$ and his reduction of $r \\in \\{25,26\\}$ to three residual orders, we eliminate those residual cases and then prove the cases $r=27,28,29$. The first structural ingredient is a Kempe-chain construction: if a $k$-critical graph has a vertex of degree $k-1$, then it contains a branch-clean essential immersion of $K_k$. Essential immersions are crossing-monotone, so a critical counterexample must have minimum degree at least $k$. For $r=27$, this one-unit degree gain, Gallai's join structure, critical-graph edge bounds, and induced-subgraph averaging close every possible order. For $r=28$ and $r=29$, the remaining near-$2r$ orders are converted to dense complements. Stehlík's coloring theorem makes the odd-order complements factor-critical; a clique-partition obstruction yields an anti-tight matching property; and Tutte barriers, Hall-type expansion, and deficit bookkeeping eliminate the final cases. At order 58 for $r=29$, Rabern's coloring inequality handles the regular case, while the last degree-deficit-two case is reduced to two disjoint triangles and a finite barrier analysis.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.04771",
  "author_total": 2,
  "authors": [
    {
      "name": "Sen Cao"
    },
    {
      "name": "Sanjit Singh Mehat"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9e6tegbznj1gh.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.04771",
  "fulltext_url": null,
  "id": "9e6tegbznj1gh",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.04771",
  "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",
      "1703|Advanced Graph Theory Research"
    ]
  ],
  "openalex": "W7211881340",
  "pdf_url": "https://arxiv.org/pdf/2609.04771",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-04",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9e6tegbznj1gh.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:18Z",
      "native_id": "2609.04771",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.04771"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Albertson's Conjecture for Chromatic Numbers at Most 29",
  "title_text": "Albertson's Conjecture for Chromatic Numbers at Most 29",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:18Z",
  "url": "https://research.draconitic.com/paper/9e6tegbznj1gh",
  "venue": "arXiv",
  "versions": []
}