Remark: Family 180. Barnette's Hamiltonian-cycle conjecture

Proves that every finite simple cubic bipartite planar $3$-vertex-connected graph has a Hamiltonian cycle, resolving Barnette's conjecture. Equivalently, every three-edge path in a finite simple cubic $3$-vertex-connected bipartite Pfaffian graph lies in a Hamiltonian cycle.

The authors are OpenAI. This companion does not verify the proof. The claim is not a theorem of the Enter series and not a premise of any registered sheet. It does not close O.1. O.1 stays NOT_CLOSED. It is not a de Sitter state and not the QdS external-leg scheme. No quantum coefficient is admitted. No compilation is asserted.