Result 180, Combinatorics

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.

Lean formalization Proof

The bigger picture

Why it matters

A Hamiltonian cycle is a route that visits every vertex exactly once and returns to its start. The unreviewed manuscript claims that specific structural conditions guarantee such a route through a planar network.

What changes?

The claimed result resolves Barnette's conjecture for finite simple graphs, meaning finitely many vertices, with no loops or repeated edges. Each vertex must have exactly three incident edges. The vertices must split into two groups with every edge joining the groups, and the graph must admit a drawing without edge crossings. It must also remain connected after deleting any one or two vertices. Under all these conditions, the manuscript asserts the existence of a Hamiltonian cycle.

What does that help mathematicians do?

For a graph meeting these assumptions, a Hamiltonian cycle would leave exactly one unused edge at each vertex. Those unused edges therefore pair up all vertices without overlap. The claim would thus guarantee a particularly structured pairing: removing its edges leaves one cycle containing every vertex, rather than several separate cycles. This connects a global visiting route to a concrete decomposition of the graph.

Are there practical applications?

Its immediate value is foundational: future arguments about this graph class could use a cycle containing every vertex as a structural starting point, rather than assume one exists. The supplied abstract states an existence theorem, not a procedure or running-time bound for finding the cycle, so it does not establish practical routing performance.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

Manuscript

Paired states and Hamiltonian cycles in cubic bipartite planar graphs

September 24, 2026 11 pages

We prove Barnette's conjecture: every finite simple cubic bipartite planar 3-vertex-connected graph has a Hamiltonian cycle.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:Paired-states-and-Hamiltonian-cycles-in-cubic-bipartite-planar-graphs-September-24-2026,
  author = {{OpenAI}},
  title = {{Paired states and Hamiltonian cycles in cubic bipartite planar graphs}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Paired-states-and-Hamiltonian-cycles-in-cubic-bipartite-planar-graphs-September-24-2026/paper.pdf}{OAI:Paired-states-and-Hamiltonian-cycles-in-cubic-bipartite-planar-graphs-September-24-2026}},
  year = {2026}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/180.md.

Barnette’s Hamiltonian-cycle conjecture

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Scope

Barnette's conjecture states that every finite simple cubic bipartite planar graph that is 3-vertex-connected has a Hamiltonian cycle. The formalization proves this statement for every such graph. The selected result is the existence of a cycle visiting every vertex exactly once.

Comparator links

Result Comparator statement
Barnette's Hamiltonian-cycle conjecture BarnetteHamiltonian.lean

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 7 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.