{"paper":{"title":"Structural description of (bull, house)-free graphs","license":"http://creativecommons.org/licenses/by/4.0/","headline":"A structural description of (bull, house)-free graphs shows that (bull, P5)-free graphs have only finitely many k-critical instances for each fixed k.","cross_cats":[],"primary_cat":"math.CO","authors_text":"Chinh T. Hoang, Manoj Belavadi","submitted_at":"2026-04-30T08:43:57Z","abstract_excerpt":"The bull is a graph consisting of a triangle and two pendant edges. The P_5 is the chordless path on five vertices. The house is the complement of a P_5. A graph is k-critical if it is k-chromatic but each of its proper induced subgraphs is (k-1)-colorable. It is known that the number of k-critical P_5-free graphs and bull-free graphs are infinite for large enough k. We give a structural description of (bull, house)-free graphs and also (bull, P_5)-free graphs. Using these structural properties we prove that for any fixed k, the number of k-critical (bull, P_5)-free graphs is finite. This impr"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We give a structural description of (bull, house)-free graphs and also (bull, P_5)-free graphs. Using these structural properties we prove that for any fixed k, the number of k-critical (bull, P_5)-free graphs is finite.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"That the claimed structural description correctly captures every (bull, house)-free graph via the decomposition or case analysis used in the proof, without missing cases that would allow infinitely many critical graphs.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Structural characterizations of (bull, house)-free and (bull, P5)-free graphs enable a finiteness proof for k-critical (bull, P5)-free graphs and a short proof of perfect divisibility.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"A structural description of (bull, house)-free graphs shows that (bull, P5)-free graphs have only finitely many k-critical instances for each fixed k.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"275685c28bc638e4395b91225fb8b34d3609721d610d7bf09415727d0949b00c"},"source":{"id":"2604.27594","kind":"arxiv","version":2},"verdict":{"id":"ceeb58f3-3198-489a-a8cb-82f20594c09f","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-07T08:53:14.708082Z","strongest_claim":"We give a structural description of (bull, house)-free graphs and also (bull, P_5)-free graphs. Using these structural properties we prove that for any fixed k, the number of k-critical (bull, P_5)-free graphs is finite.","one_line_summary":"Structural characterizations of (bull, house)-free and (bull, P5)-free graphs enable a finiteness proof for k-critical (bull, P5)-free graphs and a short proof of perfect divisibility.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"That the claimed structural description correctly captures every (bull, house)-free graph via the decomposition or case analysis used in the proof, without missing cases that would allow infinitely many critical graphs.","pith_extraction_headline":"A structural description of (bull, house)-free graphs shows that (bull, P5)-free graphs have only finitely many k-critical instances for each fixed k."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2604.27594/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"doi_compliance","ran_at":"2026-05-19T19:04:02.531144Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"3faf59f62bd8e3bf6698d107275bc8c954aca42609ad31312a5ac3dc0259e853"},"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"}