{
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  "slug": "three-coloured-paths-partial-results",
  "title": "Three-coloured paths in six-chromatic graphs: partial results",
  "shortTitle": "Partial progress on three-coloured paths",
  "url": "https://evidencepress.org/releases/three-coloured-paths-partial-results/",
  "oneLine": "A five-colouring theorem for a mixed cover class, conditional finite exclusions and explicit repair obstructions; the parent problem remains unresolved.",
  "abstract": "The dossier proves five-colourability for the union of two graphs with clique components of order at most three and one star forest, allowing overlaps. It gives a deficit-one triangle-factor result, a separate critical exclusion at order fourteen, catalogue-dependent exclusions through thirteen and selected order-fifteen exclusions. Explicit repair obstructions and a small-defect repair lemma delimit a partial-Latin-square route. The full path-colouring problem is not solved.",
  "datePublished": "2026-09-08",
  "dateModified": "2026-09-08",
  "version": "0.1.0-candidate",
  "doi": "10.5281/zenodo.22655876",
  "doiUrl": "https://doi.org/10.5281/zenodo.22655876",
  "conceptDoi": "10.5281/zenodo.22655875",
  "pdfUrl": "https://github.com/ipitchford/three-coloured-paths-partial-results/releases/download/v0.1.0-candidate/three-coloured-paths-partial-results-0.1.0-candidate.pdf",
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  "zenodoUrl": "https://zenodo.org/records/22655876",
  "repoUrl": "https://github.com/ipitchford/three-coloured-paths-partial-results",
  "releaseUrl": "https://github.com/ipitchford/three-coloured-paths-partial-results/releases/tag/v0.1.0-candidate",
  "markdownUrl": "https://evidencepress.org/releases/three-coloured-paths-partial-results/index.md",
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  "authors": [
    "Anonymous"
  ],
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  "status": "unrefereed-candidate",
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    "independentlyReproduced": false,
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    "detail": "Unrefereed partial-results candidate. AIM-COMBINATORICS-0050 remains unresolved. The minimum-order bound depends on external catalogue completeness through thirteen; only selected order-fifteen cases are closed. No external specialist validation, formal verification or priority clearance is established."
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      "label": "Availability and archiving",
      "question": "Is the evidence package publicly retrievable from an archive under a persistent identifier?",
      "state": "passed",
      "evidenceUrl": "https://zenodo.org/records/22655876",
      "note": "Public archive bytes checked before site deployment."
    },
    {
      "dimension": "internalReplay",
      "label": "Internal replay",
      "question": "Does the producer’s own pipeline reproduce the stated result from the archived package?",
      "state": "passed",
      "note": "Sixteen selected checks and clean-extraction/hostile controls. Universal claims rest on written arguments.",
      "evidenceUrl": "https://github.com/ipitchford/three-coloured-paths-partial-results/blob/v0.1.0-candidate/REPLAY_ACCEPTANCE.json"
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      "label": "Independent rerun",
      "question": "Has someone else run the supplied implementation and obtained the stated result?",
      "state": "not-assessed",
      "note": "Internal AI role review and producer replay do not establish this external assurance dimension."
    },
    {
      "dimension": "independentReimplementation",
      "label": "Independent reimplementation",
      "question": "Has someone else reached the result from an independent implementation?",
      "state": "not-assessed",
      "note": "Internal AI role review and producer replay do not establish this external assurance dimension."
    },
    {
      "dimension": "formalVerification",
      "label": "Formal verification",
      "question": "Is a formalised statement machine-checked, and over which trusted base?",
      "state": "not-assessed",
      "note": "Internal AI role review and producer replay do not establish this external assurance dimension."
    },
    {
      "dimension": "specialistReview",
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      "question": "Has a domain specialist assessed the argument?",
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      "note": "Internal AI role review and producer replay do not establish this external assurance dimension."
    },
    {
      "dimension": "editorialPeerReview",
      "label": "Editorial peer review",
      "question": "Has a journal or venue run peer review to a decision?",
      "state": "not-assessed",
      "note": "Internal AI role review and producer replay do not establish this external assurance dimension."
    },
    {
      "dimension": "dataEnvironmentReproducibility",
      "label": "Data and environment reproducibility",
      "question": "Are data and computational environment pinned well enough to rebuild?",
      "state": "partial",
      "evidenceUrl": "https://github.com/ipitchford/three-coloured-paths-partial-results/blob/v0.1.0-candidate/ENVIRONMENT.txt",
      "note": "Python standard-library replay and Linux CI; no reconstruction of every historical solver environment."
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  "provenance": {
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    "aiAssisted": true,
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    "humanRole": "Research selection, direction, supplied review and full publication authorization; scholarly creator Anonymous.",
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  "problem": {
    "name": "AIM-COMBINATORICS-0050",
    "url": "https://www.unsolvedmath.com/"
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  "metadataProvenance": "Producer-coordinated partial-results assurance is separate from canonical publication; public availability does not upgrade external validation.",
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  "keywords": [
    "graph colouring",
    "monochromatic paths",
    "triangle factors",
    "star forests",
    "partial results",
    "unrefereed candidate"
  ],
  "keyResults": [
    "Five-colourability of two bounded clique-component covers plus a star forest, allowing cover overlap.",
    "Deficit-one triangle-factor theorem with a finite residual classification.",
    "Separate order-fourteen critical exclusion and a catalogue-dependent lower-order bound.",
    "Small-defect repair lemma and explicit failures of stronger repair shortcuts."
  ],
  "reviews": [],
  "evidencePackage": "Complete written proof notes, selected finite coverage/witness replays, clean extraction, hostile controls, portable input-hash checks, stopped-search receipts and five internal editorial role reports. https://github.com/ipitchford/three-coloured-paths-partial-results/blob/v0.1.0-candidate/EDITORIAL_DECISION.md",
  "openProblems": [
    "Resolve unrestricted mixtures of stars and triangles, including arbitrary-order triangle factors.",
    "Resolve the remaining critical order-fifteen cases with 41, 42 or 43 edges.",
    "Assess the mixed-cover proof and historical priority independently.",
    "Determine whether some starting colouring always admits bipartite repair in the edge-disjoint triangle-factor subclass."
  ],
  "mathObjects": [],
  "relatedWorks": [
    {
      "citation": "Garrison (2015), Good Graph Hunting, arXiv:1508.01833; exceptional three-colour path case.",
      "url": "https://arxiv.org/abs/1508.01833",
      "doi": "10.48550/arXiv.1508.01833"
    },
    {
      "citation": "Aharoni et al. (2018), Ramsey-nice families of graphs; triangle factors and known galaxy bounds.",
      "url": "https://doi.org/10.1016/j.ejc.2018.04.007",
      "doi": "10.1016/j.ejc.2018.04.007"
    },
    {
      "citation": "Holliday, Vandenbussche and Westlund (2016), Theorem 13: degree-choosability characterization.",
      "url": "https://doi.org/10.37236/5687",
      "doi": "10.37236/5687"
    },
    {
      "citation": "Gao and Postle (2019), On the minimal edge density of K4-free 6-critical graphs.",
      "url": "https://www.columbia.edu/~wg2279/k6/k6_rev.pdf",
      "doi": null
    },
    {
      "citation": "McKay-hosted, Lalonde-generated edge-critical catalogue; completeness remains external.",
      "url": "https://users.cecs.anu.edu.au/~bdm/data/graphs.html",
      "doi": null
    }
  ],
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}