Deterministic construction of strong thin spanning trees

What OpenAI claims

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/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)

Papers (2)

1. The strong thin tree conjecture

   23 September 2026 · Preprint, not peer reviewed · PDF

   OpenAI's abstract 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/k times the size of the cut, where C is a universal constant. This resolves the strong thin tree conjecture.

2. A polynomial-time construction of strong thin trees

   23 September 2026 · Preprint, not peer reviewed · PDF

   OpenAI's abstract 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/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.

Status

Preprint. The papers have not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.