Result 131, Theoretical computer science

Rapid mixing of graph switches for every degree sequence

Resolves the simple-undirected Kannan–Tetali–Vempala conjecture: the lazy edge-switch chain mixes in $O(n^8)$ time for every graphical labeled degree sequence. The same degree-constrained graphs can also be sampled exactly uniformly by an almost-surely terminating algorithm with expected polynomial bit running time.

Lean formalization Proof

The bigger picture

Why it matters

Can we sample a network fairly while fixing how many connections each vertex has? The manuscript claims that a simple random edge-swapping process approaches uniform sampling efficiently for every feasible list of these connection counts.

What changes?

A graphical labeled degree sequence specifies each named vertex's connection count and admits at least one graph with no loops or repeated edges. The manuscript reports that, for every such sequence on n vertices, the lazy switch chain, which allows staying put and proposes degree-preserving edge swaps uniformly on four vertices, has total-variation mixing time at distance 1/4 at most 2 times n to the eighth power. This resolves the simple-undirected form of the Kannan-Tetali-Vempala conjecture as claimed.

What does that help mathematicians do?

The claimed bound rules out degree sequences that force this particular chain to take superpolynomial time to reach the stated accuracy. Researchers would no longer need additional restrictions on vertex degrees to obtain that guarantee. Separately, the manuscript reports an exactly uniform sampler for every graphical labeled degree vector. It uses unbiased random bits, terminates almost surely, and has expected polynomial bit running time, eliminating approximation bias.

Are there practical applications?

The immediate value is foundational for randomized graph algorithms: the claims provide both a universal mixing guarantee and an exact sampling method under fixed degree constraints. Such sampling supports studying what graph structure can vary when degrees are held constant. The polynomial bounds alone do not establish practical speed, and the exact sampler's runtime guarantee is in expectation.

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

Polynomial mixing of the switch chain for every graphical degree sequence

September 25, 2026 22 pages

We prove the simple-undirected form of the Kannan–Tetali–Vempala conjecture: the switch chain on simple undirected graphs mixes in polynomial time for every graphical degree sequence. For a lazy chain that proposes switches uniformly on four vertices, the total-variation mixing time at distance 1/4 is at most $2n^8$ . We also give an exactly uniform sampler for every graphical labeled degree vector. It uses unbiased random bits, terminates almost surely, and has expected polynomial bit running time.

PDF LaTeX source

Cite (BibTeX)

@misc{OAI:Polynomial-Mixing-of-the-Switch-Chain-for-Every-Graphical-Degree-Sequence-September-25-2026,
  author = {{OpenAI}},
  title = {{Polynomial mixing of the switch chain for every graphical degree sequence}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Polynomial-Mixing-of-the-Switch-Chain-for-Every-Graphical-Degree-Sequence-September-25-2026/main.pdf}{OAI:Polynomial-Mixing-of-the-Switch-Chain-for-Every-Graphical-Degree-Sequence-September-25-2026}},
  year = {2026}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/131.md.

Rapid mixing of graph switches for every degree sequence

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

Polynomial mixing of the switch chain for every graphical degree sequence

Scope

The Kannan–Tetali–Vempala conjecture asks for polynomial mixing of the switch chain for every graphical degree sequence. The formalization proves the simple undirected case: for $n\ge4$ , the lazy chain that proposes switches on four vertices has total-variation mixing time at distance $1/4$ at most $2n^8$ . It also proves switch connectivity, the stated exponential total-variation bound, and a spectral gap of at least $1/(24n^2\binom n4)$ when more than one graph realizes the degree sequence.

The paper's exactly uniform sampling algorithm is outside these selected chain estimates.

Comparator links

Result	Comparator statement
Polynomial mixing and connectivity for every graphical degree sequence	SwitchChain.lean

At a glance

Subject: Theoretical computer science
Manuscripts: 1
Pages: 22
Dated: Sep 25, 2026
Lean formalization: Yes
Reasoning summary: No

Named problems addressed

Simple-undirected Kannan–Tetali–Vempala conjectureProved

Indexed by Emergent Mind with an LLM from the text above; check the manuscripts for the precise statements.

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