{
  "id": "barnettes-conjecture",
  "title": "Barnette's conjecture",
  "description": "OpenAI's proof (2026) of Barnette's conjecture that cubic bipartite 3-connected planar graphs are Hamiltonian; through sphere maps, pairs of face-vertex matchings, a cancelling exponential sum and two induced trees in the dual triangulation.",
  "units": [
    {
      "id": "sphere-maps",
      "title": "Sphere maps",
      "reader": "sphere-maps.html",
      "source": "src/sphere-maps.md",
      "source_sha256": "b568ea8ac20341967e8362574669292ef4a0a7c78b0fbe3f2898193fa09c9327",
      "status": "author_self_checked"
    },
    {
      "id": "triangulations-from-cubic-bipartite-graphs",
      "title": "Triangulations from cubic bipartite graphs",
      "reader": "triangulations-from-cubic-bipartite-graphs.html",
      "source": "src/triangulations-from-cubic-bipartite-graphs.md",
      "source_sha256": "c3852cbab8e9c6417910d1710271a303a6e9b575c028150ad1c17e9f28e5badb",
      "status": "author_self_checked"
    },
    {
      "id": "pairs-of-states",
      "title": "Pairs of states",
      "reader": "pairs-of-states.html",
      "source": "src/pairs-of-states.md",
      "source_sha256": "d6dcbd56e41a439d4b60a76b795097b4b947f0a9978dd705844d479e43a27d31",
      "status": "author_self_checked"
    },
    {
      "id": "a-cancelling-sum-over-pairs",
      "title": "A cancelling sum over pairs",
      "reader": "a-cancelling-sum-over-pairs.html",
      "source": "src/a-cancelling-sum-over-pairs.md",
      "source_sha256": "4c32258e929e12faec119a9729aa032f20ff8366b2ab4f749a41bc56c66e403e",
      "status": "author_self_checked"
    },
    {
      "id": "two-induced-trees-and-a-hamiltonian-cycle",
      "title": "Two induced trees and a Hamiltonian cycle",
      "reader": "two-induced-trees-and-a-hamiltonian-cycle.html",
      "source": "src/two-induced-trees-and-a-hamiltonian-cycle.md",
      "source_sha256": "fd2d3f6f083882fb34790679153425a16177da67f4f1e30397fde93e69d8807a",
      "status": "author_self_checked"
    }
  ],
  "prerequisites": [],
  "status": "author_self_checked",
  "proof_providers": "providers.json",
  "provenance": "provenance.json",
  "licence": "CC0-1.0"
}