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  "slug": "theta-graphs-c3-e-positive",
  "title": "Theta graphs with a three-edge path are e-positive",
  "shortTitle": "Three paths, positive coefficients",
  "url": "https://evidencepress.org/releases/theta-graphs-c3-e-positive/",
  "oneLine": "A weighted regrouping proves the next infinite family of theta graphs is e-positive, with exact control of the cancellation boundary.",
  "abstract": "Unrefereed written-proof candidate: theta(a,b,3) is e-positive for every a>=b>=3. The proof builds on Chen, He and Wang’s composition expansion and partial-reversal method; the contribution is c=3 negative-set and full weighted-fibre accounting, with an analytic zero-group classification. Exact finite tests through 24 vertices are internal checks, not a proof of the universal quantifier, formal verification or external peer review. General shortest path at least four remains open here.",
  "datePublished": "2026-10-12",
  "dateModified": "2026-10-12",
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  "doi": "10.5281/zenodo.23312724",
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      "note": "Fresh-extraction exact replay covers 81 graphs through 24 vertices, 971165 compositions and 170224 groups, plus gap accounting, nine brute-force graph comparisons and four semantic negative controls under normal and optimized Python. Finite tests do not prove the unrestricted written theorem.",
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  "problem": {
    "name": "Wang’s theta-graph e-positivity conjecture: shortest path three",
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  "metadataProvenance": "User-supplied research and review, revised 12 October 2026; original inputs retained separately from this revised package.",
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  "keywords": [
    "chromatic symmetric functions",
    "theta graphs",
    "e-positivity",
    "compositions",
    "weighted regrouping"
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    "For all integers a>=b>=3, theta(a,b,3) is e-positive.",
    "Full weighted preimages are nonnegative even when the reversal map has collisions.",
    "Nonempty groups with first part at least two have zero sum exactly for the specified gap shape (0,2,2) with first part four and crossing part two."
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    {
      "citation": "Chen, He and Wang, Clocks are e-positive, Theorem 3.1 and partial-reversal full-fibre method; published in Discrete Mathematics 349 (2026), 114723.",
      "url": "https://arxiv.org/abs/2410.07581v2",
      "doi": "10.48550/arXiv.2410.07581"
    },
    {
      "citation": "Wang, All cycle-chords are e-positive: the shortest-path-one predecessor.",
      "url": "https://doi.org/10.1007/s00026-025-00753-2",
      "doi": "10.1007/s00026-025-00753-2"
    },
    {
      "citation": "Wang and Zhou, A composition method for neat formulas of chromatic symmetric functions.",
      "url": "https://doi.org/10.1016/j.aam.2025.102886",
      "doi": "10.1016/j.aam.2025.102886"
    },
    {
      "citation": "Stanley, A symmetric function generalization of the chromatic polynomial of a graph (1995).",
      "url": "https://doi.org/10.1006/aima.1995.1020",
      "doi": "10.1006/aima.1995.1020"
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