{"paper":{"title":"Reconstructing a graph from its Bell colouring graph","license":"http://creativecommons.org/licenses/by/4.0/","headline":"Every n-vertex graph with no vertex of degree n-1 is uniquely determined by its Bell colouring graph.","cross_cats":[],"primary_cat":"math.CO","authors_text":"Brian Hearn","submitted_at":"2026-04-14T17:40:00Z","abstract_excerpt":"The Bell colouring graph $\\mathcal{B}(G)$ of a graph $G$ is the graph whose vertices are the partitions of the vertex set of $G$ into independent sets, with an edge between two partitions if and only if one can be obtained from the other by changing the part of a single vertex of $G$. 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. 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); if this degree condition fails the reconstruction may not be unique.","pith_extraction_headline":"Every n-vertex graph with no vertex of degree n-1 is uniquely determined by its Bell colouring graph."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2604.13005/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}