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On the other hand, we prove that every planar graph $G$ contains a forest $F$ such that the Alon-Tarsi number of $G - E(F)$ is at most $3$, and hence $G - E(F)$ is 3-paintable and 3-choosable."},"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":"1906.01506","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-06-04T15:14:59Z","cross_cats_sorted":[],"title_canon_sha256":"122ed6a2bd86b26bb13c2ed8e6a1fb7620701e92e06019150c94a9af1f12bdc9","abstract_canon_sha256":"595394783e94cfff65b27c1b80825ac9411fb39e52e77b7b0d1d6f3dd522fa09"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:44:16.337754Z","signature_b64":"fd+5XERttdjuC68gu7YRHCIKbQ5bmXeIpCygSzYwlQJuqoksyjFp1iEoCTzEiOSk66HweLD2ALcUjmIB1Cu/Cg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3969035e47238084043b84600575fa46cef7eebd24dea0b908be5f721bff1577","last_reissued_at":"2026-05-17T23:44:16.337102Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:44:16.337102Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The Alon-Tarsi number of subgraphs of a planar graph","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Ringi Kim, Seog-Jin Kim, Xuding Zhu","submitted_at":"2019-06-04T15:14:59Z","abstract_excerpt":"This paper constructs a planar graph $G_1$ such that for any subgraph $H$ of $G_1$ with maximum degree $\\Delta(H) \\le 3$, $G_1-E(H)$ is not $3$-choosable, and a planar graph $G_2$ such that for any star forest $F$ in $G_2$, $G_2-E(F)$ contains a copy of $K_4$ and hence $G_2-E(F)$ is not $3$-colourable. On the other hand, we prove that every planar graph $G$ contains a forest $F$ such that the Alon-Tarsi number of $G - E(F)$ is at most $3$, and hence $G - E(F)$ is 3-paintable and 3-choosable."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1906.01506","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":""},"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":"1906.01506","created_at":"2026-05-17T23:44:16.337170+00:00"},{"alias_kind":"arxiv_version","alias_value":"1906.01506v1","created_at":"2026-05-17T23:44:16.337170+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1906.01506","created_at":"2026-05-17T23:44:16.337170+00:00"},{"alias_kind":"pith_short_12","alias_value":"HFUQGXSHEOAI","created_at":"2026-05-18T12:33:18.533446+00:00"},{"alias_kind":"pith_short_16","alias_value":"HFUQGXSHEOAIIBB3","created_at":"2026-05-18T12:33:18.533446+00:00"},{"alias_kind":"pith_short_8","alias_value":"HFUQGXSH","created_at":"2026-05-18T12:33:18.533446+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/HFUQGXSHEOAIIBB3QRQAK5P2I3","json":"https://pith.science/pith/HFUQGXSHEOAIIBB3QRQAK5P2I3.json","graph_json":"https://pith.science/api/pith-number/HFUQGXSHEOAIIBB3QRQAK5P2I3/graph.json","events_json":"https://pith.science/api/pith-number/HFUQGXSHEOAIIBB3QRQAK5P2I3/events.json","paper":"https://pith.science/paper/HFUQGXSH"},"agent_actions":{"view_html":"https://pith.science/pith/HFUQGXSHEOAIIBB3QRQAK5P2I3","download_json":"https://pith.science/pith/HFUQGXSHEOAIIBB3QRQAK5P2I3.json","view_paper":"https://pith.science/paper/HFUQGXSH","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1906.01506&json=true","fetch_graph":"https://pith.science/api/pith-number/HFUQGXSHEOAIIBB3QRQAK5P2I3/graph.json","fetch_events":"https://pith.science/api/pith-number/HFUQGXSHEOAIIBB3QRQAK5P2I3/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/HFUQGXSHEOAIIBB3QRQAK5P2I3/action/timestamp_anchor","attest_storage":"https://pith.science/pith/HFUQGXSHEOAIIBB3QRQAK5P2I3/action/storage_attestation","attest_author":"https://pith.science/pith/HFUQGXSHEOAIIBB3QRQAK5P2I3/action/author_attestation","sign_citation":"https://pith.science/pith/HFUQGXSHEOAIIBB3QRQAK5P2I3/action/citation_signature","submit_replication":"https://pith.science/pith/HFUQGXSHEOAIIBB3QRQAK5P2I3/action/replication_record"}},"created_at":"2026-05-17T23:44:16.337170+00:00","updated_at":"2026-05-17T23:44:16.337170+00:00"}