Result 174, Combinatorics

Deterministic construction of strong thin spanning trees

Resolves the strong thin-tree conjecture constructively. Every finite loopless k-edge-connected multigraph on at least two vertices has a spanning tree containing at most a universal C/kC/k fraction of the edges of every cut. Such a tree can be found deterministically in polynomial time, even with binary-encoded parallel-edge multiplicities.

Lean formalization Proof

The bigger picture

Why it matters

Can a network be reduced to a tree that connects every vertex without overusing any boundary between groups of vertices? The manuscripts claim this is always possible with a bound controlled only by the network's edge connectivity.

What changes?

The manuscripts report this for every finite loopless multigraph on at least two vertices, with edge connectivity k at least one. Parallel edges are allowed. A spanning tree connects all vertices without cycles. A cut contains the edges crossing a division of the vertices into two nonempty groups; k-edge-connectivity means every cut has at least k edges. The tree contains at most a C/k fraction of every cut's edges, simultaneously, where C is a universal constant independent of the graph and k.

What does that help mathematicians do?

The result would let researchers retain connectivity while controlling how heavily the chosen tree occupies every separation of the graph. This is stronger than controlling only the total number of retained edges or a selected family of cuts. As guaranteed edge connectivity increases, the permitted fraction decreases proportionally. No additional factor depending on the number of vertices weakens this guarantee.

Are there practical applications?

The immediate value is foundational, with an algorithmic consequence: the construction manuscript reports a deterministic algorithm running in polynomial time in the binary input length. This includes parallel-edge multiplicities encoded in binary, so the guarantee does not require listing every repeated edge separately. The construction also covers a one-vertex graph. This supplies a way to construct the promised trees, not just establish their existence.

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.

2 manuscripts

The strong thin tree conjecture

September 23, 2026 52 pages Main result formalized in Lean

We prove that every finite loopless k-edge-connected multigraph on at least two vertices, with k ≥ 1, has a spanning tree meeting each cut in at most C/kC/k times the size of the cut, where C is a universal constant. This resolves the strong thin tree conjecture.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:The-strong-thin-tree-conjecture-September-23-2026,
  author = {{OpenAI}},
  title = {{The strong thin tree conjecture}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/The-strong-thin-tree-conjecture-September-23-2026/paper.pdf}{OAI:The-strong-thin-tree-conjecture-September-23-2026}},
  year = {2026}
}

A polynomial-time construction of strong thin trees

September 23, 2026 38 pages

We give a deterministic polynomial-time construction of strong thin trees. Given a finite k-edge-connected loopless multigraph on at least one vertex, the algorithm constructs a spanning tree meeting every cut in at most a C/kC/k fraction of its edges, for a universal constant C. The running time is polynomial in the binary input length, including when parallel-edge multiplicities are encoded in binary.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:A-polynomial-time-construction-of-strong-thin-trees-September-23-2026,
  author = {{OpenAI}},
  title = {{A polynomial-time construction of strong thin trees}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/A-polynomial-time-construction-of-strong-thin-trees-September-23-2026/paper.pdf}{OAI:A-polynomial-time-construction-of-strong-thin-trees-September-23-2026}},
  year = {2026}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/174.md.

Deterministic construction of strong thin spanning trees

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

Scope

The strong thin-tree conjecture asks for spanning trees that cross every cut sparsely relative to the original graph. The formalized result gives one absolute constant C>0C>0 such that every finite loopless kk-edge-connected multigraph with at least two vertices has a spanning tree crossing each nontrivial cut at most (C/k)(C/k) times the original cut size. Parallel edges remain distinct. No numerical value of CC or construction algorithm is asserted.

The strong thin-tree problem asks for a spanning tree that crosses every cut sparsely relative to the original graph. The formalization gives one deterministic polynomial-time algorithm and an absolute constant C>0C>0 such that a finite kk-edge-connected loopless multigraph yields a spanning tree using at most a C/kC/k fraction of the edges of every cut. The input may encode parallel-edge multiplicities in binary, and the running time is polynomial in that binary input length.

Comparator links

Result Comparator statement
Strong thin-tree bound StrongThinTree.lean
Polynomial-time construction of strong thin trees AlgorithmicThinTrees.lean

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 8 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.