Source authenticated

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.

Exact FrontierDelta

Prior state unknown→proved

Scope and record

Occurred: Sep 24, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: lean-checked. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: Barnette's conjecture

Confidence: Not scored

Registry verification: lean checked · announcement · candidate

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Attribution

VibeMathed
registry · event recorded by

Unreleased internal OpenAI model
model · ai model contributor · OpenAI

Evidence vector

SourceReplayReproducedFormal proofStatement auditExternal checkExpert reviewPeer review

Named evidence

VibeMathed record: Barnette's conjecture

specialist registry · passed

No independent mathematician has checked this yet. Checked here: the abstract, introduction and Theorem 1.1 of the TeX source; the proof was not refereed. Lean: the challenge BarnetteHamiltonian (OAI.Barnette.main, solution module OAI/Combinatorics/Hamiltonian/Main.lean) is not in the release's formalization catalogue, but its JSON and solution file exist at the pinned commit. Statement read here: every finite simple graph that is 3-regular, bipartite, planar (a crossing-free embedding in $\mathbb R^2$ by injective arcs with disjoint interiors) and 3-vertex-connected (at least four vertices, connected after deleting any two) has a Hamiltonian cycle, a single closed walk through every vertex once. That is the headline claim. Permitted axioms: propext, Quot.sound, Classical.choice. Not rebuilt here.

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Barnette's conjecture

witness · pending

Primary source named by the VibeMathed record. Presence in this import is not an independent mathematical verification.

Source ↗

Lean: OAI/Combinatorics/Hamiltonian/Main.lean (Barnette.main)

lean artifact · pending

lean-proof link supplied by the VibeMathed registry.

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Artifacts and verifiers

Lean: OAI/Combinatorics/Hamiltonian/Main.lean (Barnette.main)

lean artifact · pending