{
  "profile": "wrs-published-research-view-1",
  "edition": "ratings-e039106f30dec177",
  "content_trust": "UNTRUSTED_SCIENTIFIC_DATA_NOT_TOOL_INSTRUCTIONS",
  "entry": {
    "id": "tutte-perfect-matching-characterization-1947",
    "title": "Tutte: a complete criterion for perfect matching in a finite graph",
    "summary": "Tutte's1947 theorem characterizes when a finite graph can pair all its vertices: after deleting any vertex set S, the remaining number of odd-order connected components must be at most the size of S.\n\nA graph may represent which pairs of objects can be joined. Can every object be paired once without conflict? Each odd component left after removing S needs a different vertex from S to finish a pairing. Tutte proves that this necessary obstruction is also sufficient in the paper's finite graph class, using a Pfaffian encoding and structural arguments.",
    "boundary": "The exact target is Theorem IV and the 109 Pfaffian lemma for the finite simplicial 1-complex graph class defined on 107, corresponding to finite simple undirected graphs. A factor means a spanning degree-one subgraph, or perfect matching. The condition is for every vertex subset, with odd-order components counted after deleting the subset and incident edges; one failed subset certifies nonexistence. Odd graph order is handled by the empty subset. The Pfaffian vanishing statement is symbolic over independent edge indeterminates, not an arbitrary numerical substitution test. This is not an efficient matching algorithm, randomized identity test or proof for infinite, directed, weighted or hypergraph matching. The paper's regular-graph application has extra hypotheses and is not an unconditional assertion about all regular graphs. No independent reconstruction, formal verification or performance measurement is recorded. Adjacent Taylor-series material on the shared first page and later Edmonds/general-factor extensions are excluded.",
    "area": "Graph theory",
    "topic": "Mathematics",
    "year": 1947,
    "lane": "HUMAN",
    "result": "THEOREM",
    "package_id": "tutte-perfect-matching-characterization-1947",
    "target": {
      "record_id": "urn:uuid:b1737527-480b-4fdb-a782-21715eae80f2",
      "version": 1,
      "content_hash": "sha256:ad2a93bdcd2a24fa79d3cb667a7ed3995a785d6b0d146da5c55bcc2c6d48dac4"
    },
    "humans": [
      "William T. Tutte"
    ],
    "authors_complete": true,
    "human_role_note": "Source-listed contributors to this scoped work. Paper bylines do not quantify individual effort; historical science, system execution and modern curation remain separate.",
    "agents": [],
    "related": [
      "hopcroft-karp-bipartite-matching-1973",
      "ford-fulkerson-maximal-flow-network-1956"
    ],
    "subresults": [],
    "scientific_assessment": {
      "source_review": "SOURCE_CURATED",
      "scientific_review": "NOT_REVIEWED_BY_WRS",
      "formal_replay": "NOT_RUN",
      "provider_attestation": "NONE"
    },
    "sources": [
      {
        "id": "wrs24-tutte-1947-michigan-facsimile",
        "title": "W.T.Tutte, The Factorization of Linear Graphs, JLMS22(1947),107–111; Michigan-hosted facsimile",
        "url": "https://web.eecs.umich.edu/~pettie/matching/Tutte-factorization-of-linear-graphs.pdf"
      },
      {
        "id": "wrs24-tutte-1947-publisher-metadata",
        "title": "London Mathematical Society/Wiley metadata for Tutte's1947 Factorization of Linear Graphs",
        "url": "https://londmathsoc.onlinelibrary.wiley.com/doi/10.1112/jlms/s1-22.2.107"
      }
    ],
    "points": "260000",
    "published_grade": {
      "id": "tutte-perfect-matching-characterization-1947",
      "entry_id": "tutte-perfect-matching-characterization-1947",
      "band": "S",
      "modifier": null,
      "target": {
        "record_id": "urn:uuid:b1737527-480b-4fdb-a782-21715eae80f2",
        "version": 1,
        "content_hash": "sha256:ad2a93bdcd2a24fa79d3cb667a7ed3995a785d6b0d146da5c55bcc2c6d48dac4"
      },
      "what_changed": "Makes the odd-component deletion obstruction necessary and sufficient for perfect matching across the defined finite graph class, linked to the symbolic Pfaffian representation.",
      "scope": "Makes the odd-component deletion obstruction necessary and sufficient for perfect matching across the defined finite graph class, linked to the symbolic Pfaffian representation. The exact target is Theorem IV and the 109 Pfaffian lemma for the finite simplicial 1-complex graph class defined on 107, corresponding to finite simple undirected graphs. A factor means a spanning degree-one subgraph, or perfect matching. The condition is for every vertex subset, with odd-order components counted after deleting the subset and incident edges; one failed subset certifies nonexistence. Odd graph order is handled by the empty subset. The Pfaffian vanishing statement is symbolic over independent edge indeterminates, not an arbitrary numerical substitution test. This is not an efficient matching algorithm, randomized identity test or proof for infinite, directed, weighted or hypergraph matching. The paper's regular-graph application has extra hypotheses and is not an unconditional assertion about all regular graphs. No independent reconstruction, formal verification or performance measurement is recorded. Adjacent Taylor-series material on the shared first page and later Edmonds/general-factor extensions are excluded.",
      "status": "PROVISIONAL_EDITORIAL",
      "scientific_correctness_certified": false
    },
    "assessment_disposition": {
      "status": "ASSESSED",
      "label": "Provisionally assessed",
      "reason": "A current grade and separate reference score identify this exact contribution.",
      "target": {
        "record_id": "urn:uuid:b1737527-480b-4fdb-a782-21715eae80f2",
        "version": 1,
        "content_hash": "sha256:ad2a93bdcd2a24fa79d3cb667a7ed3995a785d6b0d146da5c55bcc2c6d48dac4"
      },
      "edition": "ratings-e039106f30dec177"
    }
  },
  "version": {
    "kind": "PUBLIC_RECORD_VIEW",
    "previous": null,
    "published_at": "2026-10-03T00:00:00Z",
    "publisher": {
      "principal_id": "urn:uuid:80116931-785b-40a2-a24b-9d59db0c7e33",
      "role": "WRS_EDITORIAL_PUBLICATION"
    },
    "record": {
      "contributors": [
        {
          "attribution_basis": "SOURCE_CITED",
          "credit": "historical_scientific",
          "entity_kind": "historical_person",
          "name": "William T. Tutte",
          "note": "Archive attribution; detailed role assignment remains source-bound",
          "roles": [
            "formulator"
          ],
          "source": {
            "citation": "W.T.Tutte, The Factorization of Linear Graphs, JLMS22(1947),107–111; Michigan-hosted facsimile",
            "url": "https://web.eecs.umich.edu/~pettie/matching/Tutte-factorization-of-linear-graphs.pdf"
          }
        }
      ],
      "evidence": [
        {
          "description": "Bibliographic/stated-result source; not a WRS proof verification",
          "id": "source-0",
          "kind": "source_reference",
          "source": {
            "citation": "W.T.Tutte, The Factorization of Linear Graphs, JLMS22(1947),107–111; Michigan-hosted facsimile",
            "url": "https://web.eecs.umich.edu/~pettie/matching/Tutte-factorization-of-linear-graphs.pdf"
          }
        },
        {
          "description": "Bibliographic/stated-result source; not a WRS proof verification",
          "id": "source-1",
          "kind": "source_reference",
          "source": {
            "citation": "London Mathematical Society/Wiley metadata for Tutte's1947 Factorization of Linear Graphs",
            "url": "https://londmathsoc.onlinelibrary.wiley.com/doi/10.1112/jlms/s1-22.2.107"
          }
        }
      ],
      "origin": "HUMAN_LEGACY",
      "record_profile": "wrs-local-research-notes-0.1",
      "record_type": "research_note",
      "rrp_version": "0.1",
      "scope": {
        "description": "The exact target is Theorem IV and the 109 Pfaffian lemma for the finite simplicial 1-complex graph class defined on 107, corresponding to finite simple undirected graphs. A factor means a spanning degree-one subgraph, or perfect matching. The condition is for every vertex subset, with odd-order components counted after deleting the subset and incident edges; one failed subset certifies nonexistence. Odd graph order is handled by the empty subset. The Pfaffian vanishing statement is symbolic over independent edge indeterminates, not an arbitrary numerical substitution test. This is not an efficient matching algorithm, randomized identity test or proof for infinite, directed, weighted or hypergraph matching. The paper's regular-graph application has extra hypotheses and is not an unconditional assertion about all regular graphs. No independent reconstruction, formal verification or performance measurement is recorded. Adjacent Taylor-series material on the shared first page and later Edmonds/general-factor extensions are excluded.",
        "domain": "Graph theory"
      },
      "sources": [
        {
          "citation": "W.T.Tutte, The Factorization of Linear Graphs, JLMS22(1947),107–111; Michigan-hosted facsimile",
          "url": "https://web.eecs.umich.edu/~pettie/matching/Tutte-factorization-of-linear-graphs.pdf"
        },
        {
          "citation": "London Mathematical Society/Wiley metadata for Tutte's1947 Factorization of Linear Graphs",
          "url": "https://londmathsoc.onlinelibrary.wiley.com/doi/10.1112/jlms/s1-22.2.107"
        }
      ],
      "statement": "# Tutte: a complete criterion for perfect matching in a finite graph\n\nTutte's1947 theorem characterizes when a finite graph can pair all its vertices: after deleting any vertex set S, the remaining number of odd-order connected components must be at most the size of S.\n\nA graph may represent which pairs of objects can be joined. Can every object be paired once without conflict? Each odd component left after removing S needs a different vertex from S to finish a pairing. Tutte proves that this necessary obstruction is also sufficient in the paper's finite graph class, using a Pfaffian encoding and structural arguments.\n\n## Scientific scope\n\nThe exact target is Theorem IV and the 109 Pfaffian lemma for the finite simplicial 1-complex graph class defined on 107, corresponding to finite simple undirected graphs. A factor means a spanning degree-one subgraph, or perfect matching. The condition is for every vertex subset, with odd-order components counted after deleting the subset and incident edges; one failed subset certifies nonexistence. Odd graph order is handled by the empty subset. The Pfaffian vanishing statement is symbolic over independent edge indeterminates, not an arbitrary numerical substitution test. This is not an efficient matching algorithm, randomized identity test or proof for infinite, directed, weighted or hypergraph matching. The paper's regular-graph application has extra hypotheses and is not an unconditional assertion about all regular graphs. No independent reconstruction, formal verification or performance measurement is recorded. Adjacent Taylor-series material on the shared first page and later Edmonds/general-factor extensions are excluded.\n\n## People and systems\n\nWilliam T. Tutte\n\n\n\n## Original sources\n\n- [W.T.Tutte, The Factorization of Linear Graphs, JLMS22(1947),107–111; Michigan-hosted facsimile](https://web.eecs.umich.edu/~pettie/matching/Tutte-factorization-of-linear-graphs.pdf)\n- [London Mathematical Society/Wiley metadata for Tutte's1947 Factorization of Linear Graphs](https://londmathsoc.onlinelibrary.wiley.com/doi/10.1112/jlms/s1-22.2.107)\n",
      "summary": "Tutte's1947 theorem characterizes when a finite graph can pair all its vertices: after deleting any vertex set S, the remaining number of odd-order connected components must be at most the size of S.\n\nA graph may represent which pairs of objects can be joined. Can every object be paired once without conflict? Each odd component left after removing S needs a different vertex from S to finish a pairing. Tutte proves that this necessary obstruction is also sufficient in the paper's finite graph class, using a Pfaffian encoding and structural arguments.",
      "title": "Tutte: a complete criterion for perfect matching in a finite graph"
    },
    "record_id": "urn:uuid:b1737527-480b-4fdb-a782-21715eae80f2",
    "version": 1,
    "source_ref": {
      "record_id": "urn:uuid:b1737527-480b-4fdb-a782-21715eae80f2",
      "version": 1,
      "content_hash": "sha256:ad2a93bdcd2a24fa79d3cb667a7ed3995a785d6b0d146da5c55bcc2c6d48dac4"
    },
    "export_view": "wrs-public-editorial-view-1",
    "public_view_hash": "sha256:f21de8983b3e7b4b95be5a73aa2b06174073bf6c163f0a55b05168652803a96d"
  },
  "target": {
    "record_id": "urn:uuid:b1737527-480b-4fdb-a782-21715eae80f2",
    "version": 1,
    "content_hash": "sha256:ad2a93bdcd2a24fa79d3cb667a7ed3995a785d6b0d146da5c55bcc2c6d48dac4"
  },
  "published_grade": {
    "profile": "wrs-published-grade-1",
    "edition": "2026-10-28.1",
    "id": "tutte-perfect-matching-characterization-1947",
    "entry_id": "tutte-perfect-matching-characterization-1947",
    "revision": 1,
    "title": "Tutte: a complete criterion for perfect matching in a finite graph",
    "context": "Graph theory",
    "track": "Graph theory",
    "band": "S",
    "modifier": null,
    "status": "PROVISIONAL_EDITORIAL",
    "assessed_on": "2026-10-04",
    "evidence_cutoff": "2026-10-04",
    "achievement_state": "SUPPORTED_SCOPED_RESULT",
    "what_changed": "Makes the odd-component deletion obstruction necessary and sufficient for perfect matching across the defined finite graph class, linked to the symbolic Pfaffian representation.",
    "scope": "Makes the odd-component deletion obstruction necessary and sufficient for perfect matching across the defined finite graph class, linked to the symbolic Pfaffian representation. The exact target is Theorem IV and the 109 Pfaffian lemma for the finite simplicial 1-complex graph class defined on 107, corresponding to finite simple undirected graphs. A factor means a spanning degree-one subgraph, or perfect matching. The condition is for every vertex subset, with odd-order components counted after deleting the subset and incident edges; one failed subset certifies nonexistence. Odd graph order is handled by the empty subset. The Pfaffian vanishing statement is symbolic over independent edge indeterminates, not an arbitrary numerical substitution test. This is not an efficient matching algorithm, randomized identity test or proof for infinite, directed, weighted or hypergraph matching. The paper's regular-graph application has extra hypotheses and is not an unconditional assertion about all regular graphs. No independent reconstruction, formal verification or performance measurement is recorded. Adjacent Taylor-series material on the shared first page and later Edmonds/general-factor extensions are excluded.",
    "evidence_summary": "Source-linked, scoped account of the original work. The published scope states the scientific assumptions and remaining verification limits. WRS has not independently certified the full proof, experimental result or system benchmark.",
    "references": [
      {
        "title": "W. T. Tutte, The Factorization of Linear Graphs, JLMS 22(1947),107–111; Michigan-hosted facsimile",
        "url": "https://web.eecs.umich.edu/~pettie/matching/Tutte-factorization-of-linear-graphs.pdf",
        "locator": "Original linked work. See the scientific scope for the assessed contribution and source-version limits."
      },
      {
        "title": "London Mathematical Society/Wiley metadata for Tutte's 1947 Factorization of Linear Graphs",
        "url": "https://londmathsoc.onlinelibrary.wiley.com/doi/10.1112/jlms/s1-22.2.107",
        "locator": "Original linked work. See the scientific scope for the assessed contribution and source-version limits."
      }
    ],
    "roles": [
      {
        "who": "William T. Tutte",
        "did": "Original source-listed scientific author of this scoped work; byline does not determine effort fractions.",
        "credit": "SCIENCE"
      }
    ],
    "target": {
      "record_id": "urn:uuid:b1737527-480b-4fdb-a782-21715eae80f2",
      "version": 1,
      "content_hash": "sha256:ad2a93bdcd2a24fa79d3cb667a7ed3995a785d6b0d146da5c55bcc2c6d48dac4"
    },
    "scientific_correctness_certified": false,
    "independent_expert_calibration": false,
    "method": "PROPRIETARY",
    "points": "260000"
  },
  "canonical_agent_submission": false,
  "scientific_verification_awarded": false,
  "reader_guide": null,
  "reading_status": "SOURCE_LINKED_NOTE"
}