Barnette's conjecture

Theorem 1.1: every finite simple cubic bipartite planar 3-vertex-connected graph has a Hamiltonian cycle. In the dual triangulation (Stein; Alt-Payne-Schmidt-Wood) the paper splits the vertices into two induced trees with a prescribed facial pattern, using Tutte states, a signed exponential sum over pairs of states, and gluing along separating triangles. Corollaries: the Hamiltonian cycle can avoid any prescribed edge (a known equivalent strengthening, via Kelmans, Hertel and Gorsky-Steiner-Wiederrecht), and every three-edge path in a cubic 3-connected Pfaffian bipartite graph lies in a Hamiltonian cycle. The result is confined to the bipartite polyhedral class.

Theorem 1.1: every finite simple cubic bipartite planar 3-vertex-connected graph has a Hamiltonian cycle. In the dual triangulation (Stein; Alt-Payne-Schmidt-Wood) the paper splits the vertices into two induced trees with a prescribed facial pattern, using Tutte states, a signed exponential sum over pairs of states, and gluing along separating triangles. Corollaries: the Hamiltonian cycle can avoid any prescribed edge (a known equivalent strengthening, via Kelmans, Hertel and Gorsky-Steiner-Wiederrecht), and every three-edge path in a cubic 3-connected Pfaffian bipartite graph lies in a Hamiltonian cycle. The result is confined to the bipartite polyhedral class.