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Given a natural number $k$, the Bell $k$-colouring graph $\\mathcal{B}_k(G)$ and the upper-Bell $k$-colouring graph $\\mathcal{B}_{\\geq k}(G)$ are the induced subgraphs of $\\mathcal{B}(G)$ consisting of all partitions with at most $k$ parts and at least $k$ parts, respectively. 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We determine precisely when two finit"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"every n-vertex graph G with no vertices of degree n-1 is uniquely determined by its Bell colouring graph B(G), and by its upper-Bell colouring graph B_{≥k}(G) if k≤n-2","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The graphs are finite and the uniqueness holds precisely when there are no vertices of degree n-1 (or the degree bound for the k-coloring case); if this degree condition fails the reconstruction may not be unique.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Graphs without vertices of degree n-1 are uniquely determined by their Bell colouring graphs, which encode partitions into independent sets.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Every n-vertex graph with no vertex of degree n-1 is uniquely determined by its Bell colouring graph.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"9cf7d75216517f010be4863ea7517e4cf2a8643a084e7dd68a2f7a10c250a936"},"source":{"id":"2604.13005","kind":"arxiv","version":2},"verdict":{"id":"93c1f11e-51a5-4039-9cbc-7ca078c17c11","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-10T14:45:16.129235Z","strongest_claim":"every n-vertex graph G with no vertices of degree n-1 is uniquely determined by its Bell colouring graph B(G), and by its upper-Bell colouring graph B_{≥k}(G) if k≤n-2","one_line_summary":"Graphs without vertices of degree n-1 are uniquely determined by their Bell colouring graphs, which encode partitions into independent sets.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The graphs are finite and the uniqueness holds precisely when there are no vertices of degree n-1 (or the degree bound for the k-coloring case); 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