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  "slug": "symmetry-respecting-tree-torsors",
  "title": "An exact criterion for symmetry-respecting critical-group torsors",
  "shortTitle": "When tree symmetry admits a torsor",
  "url": "https://evidencepress.org/releases/symmetry-respecting-tree-torsors/",
  "oneLine": "An exact criterion and an all-path-count classification show when spanning trees support a symmetry-compatible critical-group action.",
  "abstract": "A cohomological criterion counts and constructs critical-group torsors compatible with every graph automorphism. For simple graphs formed by k internally disjoint paths between two terminals, for every k at least three, such a torsor exists exactly when path lengths are distinct and at least one is even. The all-even distinct family has 2 to the power k minus one, minus one torsor classes. Uniform subdivision can change existence despite preserving topology and the abstract automorphism group. Classical affine and subgroup-mark ingredients, finite decidability and previously asserted catalogue claims are not presented as new.",
  "datePublished": "2026-09-08",
  "dateModified": "2026-09-08",
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  "keywords": [
    "critical group",
    "spanning trees",
    "equivariant torsors",
    "group cohomology",
    "graph subdivision"
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  "keyResults": [
    "Simultaneous restriction-kernel criterion, construction and exact action counts.",
    "Complete classification of simple k-path banana subdivisions for every k>=3.",
    "Uniform subdivision changes existence at every cycle rank at least two; the positive class is not minor-closed.",
    "Exact class counts quantify non-uniqueness; cacti admit an orientation-independent construction."
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    {
      "citation": "AIM chip-firing workshop, Problem 8 (2013).",
      "url": "https://aimath.org/pastworkshops/chipfiringproblems.pdf",
      "doi": null
    },
    {
      "citation": "Chan, Church and Grochow, Rotor-routing and spanning trees on planar graphs (2015).",
      "url": "https://doi.org/10.1093/imrn/rnu025",
      "doi": "10.1093/imrn/rnu025"
    },
    {
      "citation": "Wagner, The critical group of a directed graph, Theorem 8.1 (2000).",
      "url": "https://arxiv.org/abs/math/0010241",
      "doi": null
    },
    {
      "citation": "Baker and Wang, The Bernardi process and torsor structures on spanning trees (2018).",
      "url": "https://doi.org/10.1093/imrn/rnx037",
      "doi": "10.1093/imrn/rnx037"
    },
    {
      "citation": "Ding, McDonough, Tóthmérész and Yuen, A consistent sandpile torsor algorithm for regular matroids (2025).",
      "url": "https://doi.org/10.1016/j.ejc.2025.104218",
      "doi": "10.1016/j.ejc.2025.104218"
    },
    {
      "citation": "Baker, Ding and Kim, regular orthogonal matroids and spanning quasi-trees (2026 preprint).",
      "url": "https://arxiv.org/abs/2501.08796v3",
      "doi": "10.48550/arXiv.2501.08796"
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