{
  "id": "erdos-gallai-cycle-decompositions",
  "title": "The Erdős–Gallai cycle decomposition theorem",
  "description": "OpenAI's proof (2026) of the Erdős–Gallai cycle decomposition conjecture: the edges of every graph with n vertices can be partitioned into at most Cn cycles and single edges. Lovász's path-and-cycle theorem, Haxell's hypergraph condition, routing in graphs with cut expansion, splitting at a scale, random vertex samples, closing paths through reservoirs, pair folding with lifting of cycles, and an induction over scales with a sublinear potential.",
  "units": [
    {
      "id": "paths-cycles-and-lovasz-decompositions",
      "title": "Paths, cycles and Lovász's decomposition theorem",
      "reader": "paths-cycles-and-lovasz-decompositions.html",
      "source": "src/paths-cycles-and-lovasz-decompositions.md",
      "source_sha256": "8c1dbbda8b67b4e564af07f1e8168b7c33f72b9e19f5b03118347cbcc4c8860e",
      "status": "author_self_checked"
    },
    {
      "id": "disjoint-representatives-in-hypergraphs",
      "title": "Disjoint representatives in hypergraphs",
      "reader": "disjoint-representatives-in-hypergraphs.html",
      "source": "src/disjoint-representatives-in-hypergraphs.md",
      "source_sha256": "4d4ccbd5ccc177686e0b4283b6f401df98165f0396122fd4c0fee33b9b2caa72",
      "status": "author_self_checked"
    },
    {
      "id": "routing-in-graphs-with-cut-expansion",
      "title": "Routing in graphs with cut expansion",
      "reader": "routing-in-graphs-with-cut-expansion.html",
      "source": "src/routing-in-graphs-with-cut-expansion.md",
      "source_sha256": "29f519f77ad02854422af845bfd195bbdef63294e14020f3580a7d6da3828080",
      "status": "author_self_checked"
    },
    {
      "id": "splitting-a-graph-at-a-scale",
      "title": "Splitting a graph at a scale",
      "reader": "splitting-a-graph-at-a-scale.html",
      "source": "src/splitting-a-graph-at-a-scale.md",
      "source_sha256": "1ac0f10bb2eb0130a56b8072a32d8c31e0abab65ebcebc9fef953fce57b4cfcb",
      "status": "author_self_checked"
    },
    {
      "id": "random-sampling-in-expanding-graphs",
      "title": "Random sampling in expanding graphs",
      "reader": "random-sampling-in-expanding-graphs.html",
      "source": "src/random-sampling-in-expanding-graphs.md",
      "source_sha256": "10fb6421e5e1b9f6a37473042b8f36812e572f62957520f74591d77d0a21cb39",
      "status": "author_self_checked"
    },
    {
      "id": "closing-paths-through-reservoirs",
      "title": "Closing paths through reservoirs",
      "reader": "closing-paths-through-reservoirs.html",
      "source": "src/closing-paths-through-reservoirs.md",
      "source_sha256": "d7c90808228cad0e3c8d62a85a3fdb460f247cf339b692679208c86d384b3243",
      "status": "author_self_checked"
    },
    {
      "id": "folding-pairs-and-lifting-cycles",
      "title": "Folding pairs and lifting cycles",
      "reader": "folding-pairs-and-lifting-cycles.html",
      "source": "src/folding-pairs-and-lifting-cycles.md",
      "source_sha256": "599e32e4a77db5c1075109f7d24721988ee92321c429ae956233d0557925132d",
      "status": "author_self_checked"
    },
    {
      "id": "the-linear-bound",
      "title": "The linear bound",
      "reader": "the-linear-bound.html",
      "source": "src/the-linear-bound.md",
      "source_sha256": "e37faf117efa8f71afaf19ccd919b86ffce0d2009f49c99ea4ae4ccc1cbd2037",
      "status": "author_self_checked"
    }
  ],
  "prerequisites": [
    "core course Probability (Grinstead and Snell, Introduction to Probability): independence, expectation and its linearity, binomial distributions; all probability spaces in the course are finite, and Markov's inequality and the Chernoff bounds used are proved in the fifth lesson",
    "basic graph terminology: degrees, paths, cycles, components, induced subgraphs"
  ],
  "status": "author_self_checked",
  "proof_providers": "providers.json",
  "provenance": "provenance.json",
  "licence": "CC0-1.0"
}