{
  "schemaVersion": "1.4",
  "slug": "txgraffiti-order48-successor",
  "title": "A 48-vertex cubic counterexample to independent domination versus minimum maximal matching",
  "shortTitle": "A 48-vertex cubic counterexample",
  "url": "https://evidencepress.org/releases/txgraffiti-order48-successor/",
  "oneLine": "An explicit connected cubic graph has minimum maximal matching number 15 but independent domination number 16, lowering the candidate counterexample-order upper bound from 50 to 48.",
  "abstract": "This anonymous, unrefereed candidate gives an explicit connected cubic graph G on 48 vertices with minimum maximal matching number mu*(G)=15 and independent domination number i(G)=16. The matching equality follows from an explicit maximal matching and the elementary lower bound 72/5. The independent-domination equality follows from an explicit 16-vertex witness and a compressed exhaustive proof tree excluding every independent dominating set of size at most 15. The dependency-free checker reports 437,188 tree nodes, the C++ generator reproduces the normalized gzip certificate byte for byte, and eight semantic corruptions are rejected under ordinary and optimized Python. This improves the public 50-vertex candidate-order upper bound to 48. It does not prove global minimality, uniqueness, priority, independent reproduction, formal verification, external specialist review, editorial peer review or scientific impact.",
  "datePublished": "2026-08-24",
  "dateModified": "2026-08-24",
  "version": "0.1.0-candidate",
  "doi": "10.5281/zenodo.22083656",
  "doiUrl": "https://doi.org/10.5281/zenodo.22083656",
  "conceptDoi": "10.5281/zenodo.22083655",
  "pdfUrl": "https://github.com/ipitchford/txgraffiti-order48-successor/releases/download/v0.1.0-candidate/txgraffiti_order48_successor_MANUSCRIPT_v0.1.0_candidate.pdf",
  "altPdfUrl": "https://zenodo.org/records/22083656/files/txgraffiti_order48_successor_MANUSCRIPT_v0.1.0_candidate.pdf?download=1",
  "altPdfLabel": null,
  "zenodoUrl": "https://zenodo.org/records/22083656",
  "repoUrl": "https://github.com/ipitchford/txgraffiti-order48-successor",
  "releaseUrl": "https://github.com/ipitchford/txgraffiti-order48-successor/releases/tag/v0.1.0-candidate",
  "markdownUrl": "https://evidencepress.org/releases/txgraffiti-order48-successor/index.md",
  "bibtexUrl": "https://evidencepress.org/releases/txgraffiti-order48-successor/cite.bib",
  "audioUrl": "https://evidencepress.org/assets/audio/txgraffiti-order48-successor.mp3?v=97b35753d0",
  "imageUrl": "https://evidencepress.org/assets/og/txgraffiti-order48-successor.png?v=1a39769e82",
  "coverArtUrl": "https://evidencepress.org/assets/art/txgraffiti-order48-successor.svg?v=634b1bedce",
  "media": [
    {
      "type": "audio",
      "url": "https://evidencepress.org/assets/audio/txgraffiti-order48-successor.mp3",
      "name": "Audio briefing — 48-vertex TxGraffiti successor",
      "description": "Plain-English OpenAI API synthetic-voice summary of the explicit graph theorem, certificate replay, parent-child relationship and assurance boundary; a communication aid, not additional mathematical evidence.",
      "transcriptUrl": "https://evidencepress.org/assets/audio/txgraffiti-order48-successor.txt"
    },
    {
      "type": "video",
      "url": "https://youtu.be/uytc3J_EZFc",
      "name": "Breaking the Bound  The 48 Vertex Counterexample",
      "description": "Evidence Press video overview of this release. This is a communication aid; it does not extend the release's claims or assurance status and is not additional research evidence."
    }
  ],
  "authors": [
    "Anonymous"
  ],
  "license": "CC0-1.0",
  "status": "unrefereed-candidate",
  "verification": {
    "peerReviewed": false,
    "independentlyReproduced": false,
    "formallyVerified": false,
    "internallyReplayed": true,
    "detail": "Anonymous, unrefereed certificate-backed candidate. The explicit finite graph theorem and order-48 upper bound pass bundled deterministic replay, mutation controls, clean extraction, cross-version CI and cross-archive byte readback. The five-role editorial reports found no unresolved critical, major or minor repair within the frozen package, but they are internal and model-mediated. No unaffiliated rerun, independently authored implementation, formal proof, external graph-theory specialist review, journal peer review, global-minimality proof, uniqueness result or priority determination is reported. The child extends the public 50-vertex release and resolves its below-50 search direction by example; it does not retract the parent or inherit its separate formula-graph and restricted-class theorems."
  },
  "assurance": [
    {
      "dimension": "availability",
      "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/22083656",
      "note": "The tagged GitHub prerelease and published Zenodo record expose five matching assets; unauthenticated downloads from both services matched the local SHA-256 ledger."
    },
    {
      "dimension": "internalReplay",
      "label": "Internal replay",
      "question": "Does the producer’s own pipeline reproduce the stated result from the archived package?",
      "state": "passed",
      "note": "Python 3.11 and 3.13 CI passed ordinary and optimized checks, generator reproduction, eight mutation controls, manifest verification and two deterministic archive builds.",
      "evidenceUrl": "https://github.com/ipitchford/txgraffiti-order48-successor/actions/runs/32746202452"
    },
    {
      "dimension": "independentRerun",
      "label": "Independent rerun",
      "question": "Has someone else run the supplied implementation and obtained the stated result?",
      "state": "not-assessed",
      "note": "No unaffiliated party has reported running the exact tagged package and certificate."
    },
    {
      "dimension": "independentReimplementation",
      "label": "Independent reimplementation",
      "question": "Has someone else reached the result from an independent implementation?",
      "state": "not-assessed",
      "note": "The checker, generator and package hardening were coordinated within one producer workflow; no separately authored implementation has reconstructed the result."
    },
    {
      "dimension": "formalVerification",
      "label": "Formal verification",
      "question": "Is a formalised statement machine-checked, and over which trusted base?",
      "state": "not-assessed",
      "note": "The release contains finite proof objects and deterministic programs but no proof-assistant formalization of the graph-theoretic theorem."
    },
    {
      "dimension": "specialistReview",
      "label": "Specialist review",
      "question": "Has a domain specialist assessed the argument?",
      "state": "not-assessed",
      "note": "The five-role reports are internal model-mediated editorial quality control, not external graph-theory specialist review."
    },
    {
      "dimension": "editorialPeerReview",
      "label": "Editorial peer review",
      "question": "Has a journal or venue run peer review to a decision?",
      "state": "not-assessed",
      "note": "No journal or comparable scholarly venue has conducted peer review to a decision."
    },
    {
      "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/txgraffiti-order48-successor/blob/v0.1.0-candidate/ENVIRONMENT.md",
      "note": "The package records standard-library Python, a C++20 compiler and gzip, supplies CI across two Python versions and avoids theorem-critical third-party Python packages, but no independently recreated container or functional environment is reported."
    }
  ],
  "provenance": {
    "aiGenerated": true,
    "aiAssisted": true,
    "generatedBy": [
      "AI systems under human direction",
      "OpenAI Codex",
      "Python standard library",
      "C++20 toolchain"
    ],
    "humanRole": "Research direction, mediation, claim-boundary decisions and publication authorization by the human creator; repository maintenance and Evidence Press publication are not scholarly authorship, and scholarly attribution remains Anonymous.",
    "disclosure": "AI systems contributed package recovery, code hardening, checking, editorial review, exposition and release engineering in one producer-coordinated workflow. Exact replay, model agreement and matching public hashes are not independent reproduction, external specialist review or editorial peer review."
  },
  "attributionBoundary": null,
  "problem": {
    "name": "Independent domination versus minimum maximal matching in regular graphs",
    "url": "https://doi.org/10.7151/dmgt.2317"
  },
  "orderedProductAverage": null,
  "corrections": [],
  "publicCorrections": [],
  "publicCorrectionIndexes": null,
  "pageStructureVersion": "2.0",
  "pageStructureVariant": "successor-parent-dependent",
  "pageStructureWaivers": [],
  "recordMaturity": "current",
  "metadataProvenance": "Authored prospectively for the reader-first schema 2.0 against immutable GitHub and Zenodo identities, the frozen internal review target and the public parent-release boundary.",
  "grandfatheredAtSchemaVersion": null,
  "keywords": [
    "graph theory",
    "cubic graphs",
    "independent domination",
    "minimum maximal matching",
    "TxGraffiti Conjecture",
    "proof-tree certificate",
    "computer-assisted mathematics",
    "unrefereed candidate"
  ],
  "keyResults": [
    "One explicit connected cubic graph on 48 vertices satisfies mu*(G)=15<16=i(G).",
    "The upper bound mu*(G)<=15 is witnessed by a 15-edge maximal matching, and the lower bound follows from the elementary 72/5 edge-domination count.",
    "An explicit independent dominating set gives i(G)<=16, while a 437,188-node proof tree excludes every independent dominating set of size at most 15.",
    "The result lowers the public candidate counterexample-order upper bound from 50 to 48 without establishing global minimality or uniqueness.",
    "The C++ generator reproduces the normalized compressed certificate byte for byte, and eight semantic mutations are rejected in normal and optimized Python."
  ],
  "reviews": [],
  "evidencePackage": "The immutable GitHub and Zenodo releases contain a deterministic source ZIP, manuscript, evidence supplement, five-role internal review bundle and checksum ledger. The package cross-checks JSON, edge-list and graph6 encodings; verifies the explicit matching and independent dominating set; reads the 437,188-node exhaustive certificate under normal and optimized Python; recompiles the C++ generator and reproduces the normalized gzip certificate byte for byte; rejects eight targeted semantic corruptions; validates claims, assurance and manifests; and replays from a clean extraction. Python 3.11 and 3.13 GitHub Actions jobs pass, and all five GitHub and Zenodo assets match local SHA-256 bytes after unauthenticated download. These establish public availability, scoped producer replay and byte identity, not independent mathematical reconstruction or peer review.",
  "openProblems": [
    "Determine whether any cubic counterexample exists on fewer than 48 vertices using an independently specified exhaustive search or lower-bound argument.",
    "Obtain an unaffiliated rerun of the exact tagged package and a separately authored checker for the graph, witnesses and independent-domination exclusion certificate.",
    "Seek external specialist review of the elementary matching argument, proof-tree semantics, source-to-encoding bridge and global scope boundary.",
    "Investigate uniqueness or multiplicity at order 48 without inferring either from the single supplied object.",
    "Clarify novelty and priority through a broader graph-database and literature search while preserving the current no-first-discovery ceiling."
  ],
  "relatedWorks": [
    {
      "citation": "Caro, Y., Davila, R., & Pepper, R. (2021). New results relating independence and matchings. Discussiones Mathematicae Graph Theory, 41, 921-935.",
      "url": "https://doi.org/10.7151/dmgt.2317"
    },
    {
      "citation": "Release contributors. (2026). TxGraffiti Conjecture 15/3 resolution: a certificate-backed candidate counterexample (Version 4.0.0-rc1).",
      "url": "https://doi.org/10.5281/zenodo.21852504"
    },
    {
      "citation": "TxGraffiti Conjecture 3 counterexample repository and public predecessor objects.",
      "url": "https://github.com/djma/TxGraffiti-conjecture3-counterexample"
    }
  ],
  "operatingModel": {
    "version": "1.0",
    "workId": "ep-work:txgraffiti-order48-successor",
    "attemptIds": [
      "ep-attempt:txgraffiti-order48-successor"
    ],
    "aims": [
      "science"
    ],
    "artifactRoles": [
      "research-output",
      "evidence-assessment",
      "method-demonstration",
      "communication"
    ],
    "lineageId": "txgraffiti-counterexample-order-programme",
    "accelerationPrimitives": [
      "certificate-first",
      "adversarial-controls",
      "research-lineage-reuse",
      "counterexample-proxy-first",
      "assurance-vector",
      "agent-readable-research-object"
    ],
    "decisionObject": {
      "type": "counterexample",
      "description": "An explicit 48-vertex connected cubic graph, matching and independent-dominating witnesses, plus an exhaustive proof-tree certificate establishing mu*(G)=15<16=i(G).",
      "scope": "A constructive order-48 counterexample and resulting upper bound on minimum cubic counterexample order; not a lower bound below 48, global minimality, uniqueness or priority."
    },
    "bottleneckTargeted": [
      "discovery",
      "assurance",
      "publication"
    ],
    "semanticBridge": {
      "state": "explicit",
      "description": "The manuscript defines both graph parameters, binds three graph encodings to one labelled object, proves the matching equality by witness and edge count, and specifies how the independent-domination proof tree exhausts all legal extensions before the checker accepts a leaf.",
      "remainingRisks": [
        "The graph, checker, generator and explanatory proof were integrated in one producer workflow and may share a source-to-semantics error.",
        "The proof tree establishes only the supplied order-48 object theorem; it does not search or exclude all smaller cubic graphs.",
        "The bounded literature audit cannot establish absolute novelty or priority.",
        "No external graph-theory specialist has reconstructed the argument from the immutable package."
      ]
    },
    "humanJudgmentGates": [
      "Check the graph-parameter definitions, maximal-matching lower bound and proof-tree exhaustiveness as mathematics rather than inferring them from successful execution.",
      "Keep the constructive order-48 upper bound separate from global minimality and uniqueness.",
      "Treat the parent as research lineage and prior public evidence, not independent confirmation of the child.",
      "Require external specialist and broader priority review before first, minimal, unique or definitive language.",
      "Confirm Anonymous scholarly attribution, component licences, immutable identifiers and release authority before publication."
    ],
    "parentLinks": [
      {
        "relation": "extends-result",
        "workId": null,
        "legacyReleaseSlug": "txgraffiti-c3-resolution",
        "externalUrl": null,
        "inheritedClaim": "The parent supplies a public 50-vertex candidate counterexample and explicitly leaves the unrestricted below-50 search open. The child supplies a different 48-vertex graph and lowers that upper bound by construction.",
        "inheritedAssuranceCeiling": "The parent is an unrefereed producer-side candidate. Shared problem framing, provenance and certificate architecture are dependency evidence, not independent confirmation, and a common defect can propagate across the lineage."
      }
    ],
    "assuranceTarget": {
      "dimensions": [
        "independentRerun",
        "independentReimplementation",
        "formalVerification",
        "specialistReview",
        "dataEnvironmentReproducibility",
        "semanticValidation",
        "noveltyAssessment",
        "priorityAssessment"
      ],
      "nextAction": "Obtain an unaffiliated exact-object rerun and separately authored checker, then have a graph-theory specialist assess the proof-tree semantics and commission an independently specified search below order 48.",
      "claimCeiling": "A producer-side anonymous unrefereed certificate-backed candidate proving one explicit order-48 cubic counterexample and an upper bound of 48; not global minimality, uniqueness, independent reproduction, formal verification, external specialist or editorial peer review, absolute priority or demonstrated workflow impact."
    },
    "impactClaims": [
      {
        "id": "txgraffiti-order48-science-no-impact",
        "aim": "science",
        "outcome": "Faster or more reliable discovery and assurance of finite graph counterexamples",
        "setting": "Certificate-first AI-assisted research on independent domination and minimum maximal matching in cubic graphs",
        "status": "NO_IMPACT_EVIDENCE",
        "designClass": "none",
        "comparator": "No matched conventional research, theorem-review or publication workflow was registered.",
        "estimand": "No effect on discovery time, active human effort, compute, error rate, assurance time, reuse, uptake or citation was estimated.",
        "evidenceRefs": [],
        "registeredDesignRef": null,
        "independentAssessment": null
      }
    ]
  }
}