{"paper":{"title":"On $k$-colorability of $(bull, H)$-free graphs","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Ingo Schiermeyer, Magdalena Prorok, Monika Pil\\'sniak, Nadzieja Hodur","submitted_at":"2025-09-01T18:17:45Z","abstract_excerpt":"The $3$-colorability problem is a well-known NP-complete problem and it remains NP-complete for $bull$-free graphs, where a $bull$ is the graph consisting of a $K_3$ with two pendant edges attached to two of its vertices. In this paper, for $k\\geq3$, we characterize all $k$-colorable $(bull,claw)$-free graphs containing an induced cycle of length at least $6$. Moreover, we present the full characterization of all non $4$-colorable connected $(bull,claw)$-free graphs and $(bull,chair, C_5)$-free graphs, and all non $5$-colorable connected $(bull, claw, C_5)$-free graphs."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2509.01698","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2509.01698/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"}