bibkey: kirillov2026loopy authors: Anatol Kirillov, Gleb Nenashev, Boris Shapiro, and Arkady Vaintrob year: 2026 title: “The Loopy Polynomial: from Tutte’s Universal V-Function to Bizonotopal Geometry” doi: null url: https://arxiv.org/abs/2609.07728v1 claim: “Definition 1.1 and Proposition 2.1 give the ordinary Loopy recursion and its choice independence; Problem 11.4 asks whether it determines the degree sequence for arbitrary graphs with loops included.” strata_touched:

• D5/S3/Factorization/Combinatorics/LoopyEvaluator
• D5/S3/Factorization/Combinatorics/LoopyDegreeSequence license: citation-only triage: anchor

The ordinary Loopy polynomial and degree sequences with loops

The cited source is arXiv:2609.07728v1, published September 7, 2026. Definition 1.1 gives deletion-contraction for a nonloop, multiplication under disjoint union and the looped one-vertex base value. Proposition 2.1 proves that the recursion is independent of the choices used to compute it. The repository’s labelled evaluator refines these laws to finite vertex sets, lists of edge occurrences and an accumulated-loop function. Lists retain parallel edges; contraction of a selected nonloop keeps that occurrence as one accumulated loop.

Section 6.5, including Remark 6.17 and Lemma 6.18, supplies refined-degree context. It does not prove the repository’s substitution x_r = [2r+1], nor does it prove that the ordinary Loopy polynomial recovers degrees when loops are present. The paper distinguishes the refined invariant from the ordinary one and states the latter question explicitly as Problem 11.4:

Does L_G determine the degree sequence of an arbitrary graph G, loops included?

The repository proves the equality-kernel form: for any two independently encoded finite undirected multigraphs, validity and equality of their ordinary Loopy polynomials imply equality of their complete actual degree multisets. There is no equal-order premise. Parallel edges, loops counted twice, isolates, disconnected graphs and empty graphs are all included.

Verified locator

The caller read https://arxiv.org/abs/2609.07728v1 and the public arXiv HTML at Definition 1.1, Proposition 2.1, section 6.5 (Remark 6.17 and Lemma 6.18) and Problem 11.4. Its SHA-256 is e38f48ed00c59fd2a80f59395a3b08c6a42ea0ead467ac5cce029b3886394332. The downloaded PDF has SHA-256 f3d4ae846384717ecce1c31261f021f172dad47443c7dd635d1d15b64ddd8f12. The caller read the title on PDF page 1, the ordinary/refined distinction on PDF pages 33–34, and the exact question on PDF page 63. PDF text extraction drops the refined-L typography, so the corresponding HTML MathML and alt text supply that distinction. The metadata source lists Anatol Kirillov, Gleb Nenashev, Boris Shapiro and Arkady Vaintrob, category math.CO, and publication time 2026-09-07T16:36:46Z.

The supplied bounded search covered source v1 and an exact-phrase arXiv query; supplementary Crossref, Semantic Scholar, Google and Bing responses did not establish absence of prior work. The source itself presents Problem 11.4 as open in v1. Worldwide priority remains ASSUMED-UNVERIFIED; this note makes no global novelty claim.