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Math-2399 · mathematics · Combinatorics

Enumeration of Graphs with Prescribed Degree Sequences

Status Partially solved · unsolvedmath.comLLM status Open unverified

Statement

Definitions and mathematical statement

Shared notation.
$\mathbb N$, $\mathbb Z$, $\mathbb Q$, $\mathbb R$, and $\mathbb C$ denote the natural numbers, integers, rationals, reals, and complex numbers, respectively. All additional parameters and hypotheses are specified in the question.
Statement recorded in the supplied source.
Determine, exactly or asymptotically, the number of simple labelled graphs with a prescribed nonnegative degree sequence $\mathbf d=(d_1,\ldots,d_n)$.

Formulation references: [S1], [S2].

Short English statement

Determine, exactly or asymptotically, the number of simple labelled graphs with a prescribed nonnegative degree sequence $\mathbf d=(d_1,\ldots,d_n)$.

Sources

Catalogue entry

Statement licence: CC-BY-SA-4.0 · Source file

Contributions 1

Math-2399-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2399

Polynomial mixing supports randomized relative approximation and sampling. The exact enumeration and general asymptotic counting formula asked in this record require more information than a mixing-time bound.

Enumeration of Graphs with Prescribed Degree Sequences

OpenAI's claim

OpenAI presents the following claims relevant to this question, with manuscript numbers following paper order in Sources. In manuscript 1 (family 131; 2026-09-25), the asserted result is that the lazy simple-undirected switch chain mixes for every graphical degree sequence in at most $2n^8$ steps at distance $1/4$, with an exact expected-polynomial sampler.

Mathematical statement of the claimed result

Manuscript 1.
Let $d$ be any graphical degree vector on $[n]$, $n\ge4$, and $\Omega_d$ all simple labeled graphs with that degree vector. The chain stays with probability $1/2$; otherwise it uniformly chooses four vertices and one of six ordered pairs of different perfect matchings on them, replacing the first by the second only when its two edges are present and both new edges absent. With uniform stationary law $\pi_d$, the claimed bounds are \[ \min\{t:\max_G\|P_d^t(G,\cdot)-\pi_d\|_{\rm TV}\le1/4\}\le2n^8, \qquad\operatorname{gap}(P_d)\ge[24n^2\binom n4]^{-1} \] when $|\Omega_d|>1$. A further claimed sampler returns an exactly uniform graph from $\Omega_d$, using unbiased bits, terminating almost surely with expected bit cost polynomial in binary encoding length of $d$.

Outline of the argument

For manuscript 1 (family 131; 2026-09-25), pair resampling operators are compared through triple and disjoint-pair variance estimates. A global operator inequality gives the spectral gap, and decomposition into individual switches transfers it to the specified lazy chain; exact residual correction yields uniform sampling.

Formal verification

For manuscript 1 (family 131; 2026-09-25), the documented Lean scope is connectivity, spectral-gap control, and the stated lazy-switch total-variation bound for graphical degree sequences; the exact uniform sampler is not part of those selected estimates.

This coverage does not establish a formal proof of the complete problem as stated here.

The linked family Comparator checks are SwitchChain.lean.

Their named declarations include OAI.Problem315.switch_chain_main, OAI.Problem315.switch_chain_tv_bound, OAI.Problem315.switch_connectivity.

This site has not executed the Lean code. Described statement coverage is not an independent proof check.

Status caveat

The manuscripts were produced by an internal OpenAI model. OpenAI cautions that results without formalization can contain mistakes. This site has not independently checked the proofs; the label records the scope of OpenAI's claim.

Sources

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