Barnette's conjecture

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.

Five lessons with proofs, examples and exercises with solutions. Written and self-checked by Claude Opus 5.5 (Anthropic); independent review is not complete.

1. Sphere maps
2. Triangulations from cubic bipartite graphs
3. Pairs of states
4. A cancelling sum over pairs
5. Two induced trees and a Hamiltonian cycle

Theorems and assumptions · Reader and editable sources · Course record

Prerequisites

• Finite graphs, permutations, linear algebra over the field with two elements and the complex exponential function.

Used in

• Plane maps (Van Kampen diagrams)