{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R1823",
  "content_hash": null,
  "slug": "sabidussi-claim-proof-2026",
  "type": "claim",
  "title": "A compatible circuit decomposition is claimed for every eligible Euler tour",
  "summary": "The 2026 preprint reduces the graph problem to a four-colouring theorem for gaps of a cyclic word and derives a compatible circuit decomposition.",
  "relevance": "This is the source author's claimed resolution of the canonical conjecture, currently awaiting independent external review.",
  "relevance_source": "recorded",
  "body": "Orient every edge in the direction of the given Euler tour and record the vertex at each passage. The resulting cyclic word contains each vertex degree divided by two times, hence at least twice. The paper's central cyclic-word theorem colours the gaps by the four elements of F_2 squared. Adjacent gaps receive different colours, and at each letter every colour occurs an even number of times among the two gap incidences at its occurrences.\n\nTransfer each gap colour to the corresponding tour edge. Different colours on adjacent gaps mean that the two edges in every tour transition have different colours, including the transition between the final and first edges. The even-incidence condition says that each colour class induces an even subgraph at every vertex. Every finite even multigraph splits into connected 2-regular circuits by taking an Euler tour in each nonempty component and splitting closed trails at repeated vertices. Apply this decomposition separately to the four colour classes. The resulting circuits partition all edges, and no circuit can contain both edges of a prescribed transition because each circuit is monochromatic while every transition has two colours.\n\nThe cyclic-word theorem is proved with two parity lemmas over F_2. Passing from gap colours to their nonzero successive differences gives three possible local zero-sum patterns at each letter. A global balancing lemma selects one pattern per letter so that the remaining quadratic parity conditions vanish, after which prefix sums recover the required gap colours. This establishes the stronger four-colouring statement and the stated circuit consequence. Independent external mathematical review of this new proof was not identified in the dated search, so TheoremDB presents it as review pending.",
  "status": "supported",
  "evidence_grade": "sourced",
  "scope": {
    "kind": "universal",
    "statement": "all finite connected Eulerian multigraphs with minimum degree at least four, including loops and parallel edges, with a specified Euler tour"
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://arxiv.org/abs/2607.13225v1",
      "locator": "Abstract and Sections 1-4, especially Theorems 1.1 and 2.1"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://arxiv.org/abs/2607.13225v1",
    "locator": "Abstract and Sections 1-4, especially Theorems 1.1 and 2.1"
  },
  "relations": [
    {
      "slug": "R1824",
      "title": "Pinned Lean proof in sabidussi-lean",
      "object_type": "formalization",
      "relation": "formalizes",
      "direction": "incoming"
    },
    {
      "slug": "sabidussi-compatibility-conjecture",
      "title": "sabidussi compatibility conjecture",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}