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Symmetric Graphs have symmetric Matchings

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Paper breakdownThe authors' claims, conditions and measurements, as JSON Signed receiptProof this review has not been altered
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Jan Fricke

Group Theory · Submitted 25 Jul 2016

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim Surprisingly, the reversed is also true for amenable groups: if there is a perfect matching on the graph, there is also a perfect matching on the factor graph, i. e. a group invariant ("symmetric") perfect matching on the graph.

how it is stated · asserted 1.00 read the record →

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arXiv ID1607.07426 (version 1)
Pith IDT44JSNQD, Pith's permanent record of this paper and its review
CitationJan Fricke. "Symmetric Graphs have symmetric Matchings." arXiv:1607.07426 (2016). https://pith.science/paper/T44JSNQD
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The compiled record

The paper's own sentences, arranged. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 2,942 input tokens.

What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.51 · human participants 0.01 · controlled comparison 0.03 · randomized assignment 0.01 · states quantitative result 0.03 · compares with prior results 0.30 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

Assume that there is a free group action of automorphisms on a bipartite graph. If there is a perfect matching on the factor graph, then obviously there is a perfect matching on the graph. Surprisingly, the reversed is also true for amenable groups: if there is a perfect matching on the graph, there is also a perfect matching on the factor graph, i. e. a group invariant ("symmetric") perfect matching on the graph.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 Assume that there is a free group action of automorphisms on a bipartite graph. background 0.98 0.09 0.02 asserted 0.56
s1 If there is a perfect matching on the factor graph, then obviously there is a perfect matching on the graph. background 0.93 0.13 0.05 asserted 1.00
s2 Surprisingly, the reversed is also true for amenable groups: if there is a perfect matching on the graph, there is also a perfect matching on the factor graph, i. e. a group invariant ("symmetric") perfect matching on the graph. result 0.97 0.72 0.91 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.97.

What was examined, and what was not

sources read
  • arXiv listing abstract v1
not examined
  • full text
  • figures and tables
  • references and citing sentences
  • relations to other papers
levels run
  • P abstract
not run
  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper
Original abstract

Assume that there is a free group action of automorphisms on a bipartite graph. If there is a perfect matching on the factor graph, then obviously there is a perfect matching on the graph. Surprisingly, the reversed is also true for amenable groups: if there is a perfect matching on the graph, there is also a perfect matching on the factor graph, i. e. a group invariant ("symmetric") perfect matching on the graph.

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