{
  "abstract": "A coloring of the edges of a complete graph with $r$ colors is balanced if the edges among any $r+1$ vertices use all $r$ colors. Erdős and Gyárfás (1999) conjectured that for all $r\\ge3$ the complete graph $K_{r^2+1}$ has no balanced $r$-coloring. They proved the conjecture for $r=3$ and $r=4$. We prove the conjecture for $r=5$, the first new case settled in over two decades. There is no balanced $5$-coloring of $K_{26}$, and since $K_{25}$ has one, $K_{25}$ is the largest complete graph admitting a balanced $5$-coloring. We design a reduce--dispatch--enumerate workflow, where every machine step produces an artifact that can be independently checked. First, we use a machine-checked (Lean 4) proof to reduce the conjecture to a single extremal statement about the color class with the fewest edges: no graph on $26$ vertices has between $55$ and $65$ edges while every $6$ of its vertices span between $1$ and $11$ edges. We then search on the sparse class instead of its dense complement, which makes enumeration up to isomorphism feasible. With classical stability theorems for $K_6$-free graphs, we dispatch the five near-extremal edge counts analytically. For the exhaustive enumeration on the remaining six edge counts, we design efficient encodings with SAT modulo symmetries and parallel cube-and-conquer. We also certify all eleven edge counts: each of the $12{,}296$ subproblems carries a clausal refutation checked by the independent verifier drat-trim, and all $273{,}503{,}938$ symmetry-breaking clauses are also validated against the vertex permutations that justify them.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.24484",
  "author_total": 2,
  "authors": [
    {
      "name": "Hai Xia"
    },
    {
      "name": "Stefan Szeider"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dgmyrezrnatd2.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.24484",
  "fulltext_url": null,
  "id": "dgmyrezrnatd2",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.24484",
  "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.DM"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "2607|Limits and Structures in Graph Theory"
    ]
  ],
  "openalex": "W7223041821",
  "pdf_url": "https://arxiv.org/pdf/2610.24484",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-17",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dgmyrezrnatd2.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:57Z",
      "native_id": "2610.24484",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.24484"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Erdős-Gyárfás Balanced Five-Colorings",
  "title_text": "Erdős-Gyárfás Balanced Five-Colorings",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:44Z",
  "url": "https://research.draconitic.com/paper/dgmyrezrnatd2",
  "venue": "arXiv",
  "versions": []
}