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In particular, $(k,0,k)$-choosable means $k$-colorable, $(k,0,+\\infty)$-choosable means $k$-choosable and $(k,d,+\\infty)$-choosable means $d$-defective $k$-choosable. This paper proves that there are 1-defective 3-choosable graphs that are not 4-choosable, and for any p"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2306.11995","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2023-06-21T03:12:45Z","cross_cats_sorted":[],"title_canon_sha256":"4a358febf9131fc42f91d3fe9c5f96732dbd5e0b74e86be6d5bb81d4da964c12","abstract_canon_sha256":"32135947a56e862ff6957ddb3b9dd44e390009b7b912fca52646845f870f5b16"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T06:24:01.161092Z","signature_b64":"irNaJZQeo81qFR1VzhFwSytBBWbkEX45mdLBSCOJH1wOCbotD8Nv/x5TYKCxF3jGqMZvKOGgfo0bnnlNfnfNDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"fd1ba7cfb22577a9bd37a992250ab273e56b3f5dd71c0ad6b79d2629285a7ad7","last_reissued_at":"2026-07-05T06:24:01.160740Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T06:24:01.160740Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On Two problems of defective choosability","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Jie Ma, Rongxing Xu, Xuding Zhu","submitted_at":"2023-06-21T03:12:45Z","abstract_excerpt":"Given positive integers $p \\ge k$, and a non-negative integer $d$, we say a graph $G$ is $(k,d,p)$-choosable if for every list assignment $L$ with $|L(v)|\\geq k$ for each $v \\in V(G)$ and $|\\bigcup_{v\\in V(G)}L(v)| \\leq p$, there exists an $L$-coloring of $G$ such that each monochromatic subgraph has maximum degree at most $d$. In particular, $(k,0,k)$-choosable means $k$-colorable, $(k,0,+\\infty)$-choosable means $k$-choosable and $(k,d,+\\infty)$-choosable means $d$-defective $k$-choosable. This paper proves that there are 1-defective 3-choosable graphs that are not 4-choosable, and for any p"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2306.11995","kind":"arxiv","version":2},"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/2306.11995/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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2306.11995","created_at":"2026-07-05T06:24:01.160796+00:00"},{"alias_kind":"arxiv_version","alias_value":"2306.11995v2","created_at":"2026-07-05T06:24:01.160796+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2306.11995","created_at":"2026-07-05T06:24:01.160796+00:00"},{"alias_kind":"pith_short_12","alias_value":"7UN2PT5SEV32","created_at":"2026-07-05T06:24:01.160796+00:00"},{"alias_kind":"pith_short_16","alias_value":"7UN2PT5SEV32TPJX","created_at":"2026-07-05T06:24:01.160796+00:00"},{"alias_kind":"pith_short_8","alias_value":"7UN2PT5S","created_at":"2026-07-05T06:24:01.160796+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/7UN2PT5SEV32TPJXVGJCKCVSOP","json":"https://pith.science/pith/7UN2PT5SEV32TPJXVGJCKCVSOP.json","graph_json":"https://pith.science/api/pith-number/7UN2PT5SEV32TPJXVGJCKCVSOP/graph.json","events_json":"https://pith.science/api/pith-number/7UN2PT5SEV32TPJXVGJCKCVSOP/events.json","paper":"https://pith.science/paper/7UN2PT5S"},"agent_actions":{"view_html":"https://pith.science/pith/7UN2PT5SEV32TPJXVGJCKCVSOP","download_json":"https://pith.science/pith/7UN2PT5SEV32TPJXVGJCKCVSOP.json","view_paper":"https://pith.science/paper/7UN2PT5S","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2306.11995&json=true","fetch_graph":"https://pith.science/api/pith-number/7UN2PT5SEV32TPJXVGJCKCVSOP/graph.json","fetch_events":"https://pith.science/api/pith-number/7UN2PT5SEV32TPJXVGJCKCVSOP/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/7UN2PT5SEV32TPJXVGJCKCVSOP/action/timestamp_anchor","attest_storage":"https://pith.science/pith/7UN2PT5SEV32TPJXVGJCKCVSOP/action/storage_attestation","attest_author":"https://pith.science/pith/7UN2PT5SEV32TPJXVGJCKCVSOP/action/author_attestation","sign_citation":"https://pith.science/pith/7UN2PT5SEV32TPJXVGJCKCVSOP/action/citation_signature","submit_replication":"https://pith.science/pith/7UN2PT5SEV32TPJXVGJCKCVSOP/action/replication_record"}},"created_at":"2026-07-05T06:24:01.160796+00:00","updated_at":"2026-07-05T06:24:01.160796+00:00"}