{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:U4H6HPEII6FVEK2PYDSUUZXQ5I","short_pith_number":"pith:U4H6HPEI","canonical_record":{"source":{"id":"1811.12012","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-11-29T08:33:06Z","cross_cats_sorted":[],"title_canon_sha256":"79c7030436ae0ba72d89e22077c9227c452e784efc9fff0475c821392a8a8f68","abstract_canon_sha256":"72ac490a3f0579b01c978f0b8d9936dfa03e53198ceb2bfaa93a841df7257eb8"},"schema_version":"1.0"},"canonical_sha256":"a70fe3bc88478b522b4fc0e54a66f0ea2f8aa80a9763a061afb6fffeb8e726eb","source":{"kind":"arxiv","id":"1811.12012","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1811.12012","created_at":"2026-05-17T23:59:34Z"},{"alias_kind":"arxiv_version","alias_value":"1811.12012v1","created_at":"2026-05-17T23:59:34Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1811.12012","created_at":"2026-05-17T23:59:34Z"},{"alias_kind":"pith_short_12","alias_value":"U4H6HPEII6FV","created_at":"2026-05-18T12:32:56Z"},{"alias_kind":"pith_short_16","alias_value":"U4H6HPEII6FVEK2P","created_at":"2026-05-18T12:32:56Z"},{"alias_kind":"pith_short_8","alias_value":"U4H6HPEI","created_at":"2026-05-18T12:32:56Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:U4H6HPEII6FVEK2PYDSUUZXQ5I","target":"record","payload":{"canonical_record":{"source":{"id":"1811.12012","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-11-29T08:33:06Z","cross_cats_sorted":[],"title_canon_sha256":"79c7030436ae0ba72d89e22077c9227c452e784efc9fff0475c821392a8a8f68","abstract_canon_sha256":"72ac490a3f0579b01c978f0b8d9936dfa03e53198ceb2bfaa93a841df7257eb8"},"schema_version":"1.0"},"canonical_sha256":"a70fe3bc88478b522b4fc0e54a66f0ea2f8aa80a9763a061afb6fffeb8e726eb","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:59:34.709197Z","signature_b64":"p11kQU1aBD2eZDhOFl4olEPd6w/bxQ/GcwVvd2lWLiY6AWpnqRqz6LDMMKMFmo+AXYvqqIpqOxeowrB4DyjEAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a70fe3bc88478b522b4fc0e54a66f0ea2f8aa80a9763a061afb6fffeb8e726eb","last_reissued_at":"2026-05-17T23:59:34.708625Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:59:34.708625Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1811.12012","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-17T23:59:34Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"BZG4B1pACzJ86ympFoz0t3D93WuHz3EOIFN4vKnkbSA9ljQVyfwYbx4F46i1TlNYGv2LBiuE7e1mVtUVPhBvDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T08:44:38.109475Z"},"content_sha256":"d40d410f09402d17f4cb7d44d43a991b0e8170cce90da66866f61da97ea439b2","schema_version":"1.0","event_id":"sha256:d40d410f09402d17f4cb7d44d43a991b0e8170cce90da66866f61da97ea439b2"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:U4H6HPEII6FVEK2PYDSUUZXQ5I","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The Alon-Tarsi number of a planar graph minus a matching","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Jaros{\\l}aw Grytczuk, Xuding Zhu","submitted_at":"2018-11-29T08:33:06Z","abstract_excerpt":"This paper proves that every planar graph $G$ contains a matching $M$ such that the Alon-Tarsi number of $G-M$ is at most $4$. As a consequence, $G-M$ is $4$-paintable, and hence $G$ itself is $1$-defective $4$-paintable. This improves a result of Cushing and Kierstead [Planar Graphs are 1-relaxed, 4-choosable, {\\em European Journal of Combinatorics} 31(2010),1385-1397], who proved that every planar graph is $1$-defective $4$-choosable."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1811.12012","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-17T23:59:34Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"/YAzA2+GlWyA6yeJv5ZPrbUeTAKDUPyYP0gD30WxtD/DQxRFqlByaWxp7gJmF5eWry3kctjvvBahjaz3vOm+DQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T08:44:38.109823Z"},"content_sha256":"9c23f0cd616acd833aa7ca1e1a230514b9fa723e0a3602c636bb3a3dc6ce7138","schema_version":"1.0","event_id":"sha256:9c23f0cd616acd833aa7ca1e1a230514b9fa723e0a3602c636bb3a3dc6ce7138"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/U4H6HPEII6FVEK2PYDSUUZXQ5I/bundle.json","state_url":"https://pith.science/pith/U4H6HPEII6FVEK2PYDSUUZXQ5I/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/U4H6HPEII6FVEK2PYDSUUZXQ5I/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-05-31T08:44:38Z","links":{"resolver":"https://pith.science/pith/U4H6HPEII6FVEK2PYDSUUZXQ5I","bundle":"https://pith.science/pith/U4H6HPEII6FVEK2PYDSUUZXQ5I/bundle.json","state":"https://pith.science/pith/U4H6HPEII6FVEK2PYDSUUZXQ5I/state.json","well_known_bundle":"https://pith.science/.well-known/pith/U4H6HPEII6FVEK2PYDSUUZXQ5I/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:U4H6HPEII6FVEK2PYDSUUZXQ5I","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"72ac490a3f0579b01c978f0b8d9936dfa03e53198ceb2bfaa93a841df7257eb8","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-11-29T08:33:06Z","title_canon_sha256":"79c7030436ae0ba72d89e22077c9227c452e784efc9fff0475c821392a8a8f68"},"schema_version":"1.0","source":{"id":"1811.12012","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1811.12012","created_at":"2026-05-17T23:59:34Z"},{"alias_kind":"arxiv_version","alias_value":"1811.12012v1","created_at":"2026-05-17T23:59:34Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1811.12012","created_at":"2026-05-17T23:59:34Z"},{"alias_kind":"pith_short_12","alias_value":"U4H6HPEII6FV","created_at":"2026-05-18T12:32:56Z"},{"alias_kind":"pith_short_16","alias_value":"U4H6HPEII6FVEK2P","created_at":"2026-05-18T12:32:56Z"},{"alias_kind":"pith_short_8","alias_value":"U4H6HPEI","created_at":"2026-05-18T12:32:56Z"}],"graph_snapshots":[{"event_id":"sha256:9c23f0cd616acd833aa7ca1e1a230514b9fa723e0a3602c636bb3a3dc6ce7138","target":"graph","created_at":"2026-05-17T23:59:34Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"This paper proves that every planar graph $G$ contains a matching $M$ such that the Alon-Tarsi number of $G-M$ is at most $4$. As a consequence, $G-M$ is $4$-paintable, and hence $G$ itself is $1$-defective $4$-paintable. This improves a result of Cushing and Kierstead [Planar Graphs are 1-relaxed, 4-choosable, {\\em European Journal of Combinatorics} 31(2010),1385-1397], who proved that every planar graph is $1$-defective $4$-choosable.","authors_text":"Jaros{\\l}aw Grytczuk, Xuding Zhu","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-11-29T08:33:06Z","title":"The Alon-Tarsi number of a planar graph minus a matching"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1811.12012","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:d40d410f09402d17f4cb7d44d43a991b0e8170cce90da66866f61da97ea439b2","target":"record","created_at":"2026-05-17T23:59:34Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"72ac490a3f0579b01c978f0b8d9936dfa03e53198ceb2bfaa93a841df7257eb8","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-11-29T08:33:06Z","title_canon_sha256":"79c7030436ae0ba72d89e22077c9227c452e784efc9fff0475c821392a8a8f68"},"schema_version":"1.0","source":{"id":"1811.12012","kind":"arxiv","version":1}},"canonical_sha256":"a70fe3bc88478b522b4fc0e54a66f0ea2f8aa80a9763a061afb6fffeb8e726eb","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a70fe3bc88478b522b4fc0e54a66f0ea2f8aa80a9763a061afb6fffeb8e726eb","first_computed_at":"2026-05-17T23:59:34.708625Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:59:34.708625Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"p11kQU1aBD2eZDhOFl4olEPd6w/bxQ/GcwVvd2lWLiY6AWpnqRqz6LDMMKMFmo+AXYvqqIpqOxeowrB4DyjEAw==","signature_status":"signed_v1","signed_at":"2026-05-17T23:59:34.709197Z","signed_message":"canonical_sha256_bytes"},"source_id":"1811.12012","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:d40d410f09402d17f4cb7d44d43a991b0e8170cce90da66866f61da97ea439b2","sha256:9c23f0cd616acd833aa7ca1e1a230514b9fa723e0a3602c636bb3a3dc6ce7138"],"state_sha256":"bb61210a62cc493bbc47db4aa422cbb274153ed495e7f7fdbf87a9faa2ae965e"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"n2yKj8dJfBqpElInBVJsXKAsbroXI7N72URC3uGysaMWEwVO9Ud6OqYRY4+WvlSi02KzCK435Oh95x0x1jXVAw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-31T08:44:38.111840Z","bundle_sha256":"1e56f18c1df522e71aebd4aa18643f4aae62bf78ced33fbb9a0df7d5bf31acf8"}}