{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:O6UNPP2HDBULH3KIVNJN72CW5J","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":"e996199800cc3010d49f61a5ee45c88d9a9f1df52b3093a4a0c924f70f88b79f","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-10-25T13:38:38Z","title_canon_sha256":"980284b031b81ec5da190e1797aa15ac7b6029c60848b56766d8a8670114de10"},"schema_version":"1.0","source":{"id":"1210.6831","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1210.6831","created_at":"2026-05-18T03:42:23Z"},{"alias_kind":"arxiv_version","alias_value":"1210.6831v1","created_at":"2026-05-18T03:42:23Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1210.6831","created_at":"2026-05-18T03:42:23Z"},{"alias_kind":"pith_short_12","alias_value":"O6UNPP2HDBUL","created_at":"2026-05-18T12:27:16Z"},{"alias_kind":"pith_short_16","alias_value":"O6UNPP2HDBULH3KI","created_at":"2026-05-18T12:27:16Z"},{"alias_kind":"pith_short_8","alias_value":"O6UNPP2H","created_at":"2026-05-18T12:27:16Z"}],"graph_snapshots":[{"event_id":"sha256:ab96ec671d18386753ec4657e5920c8c0bd9cd9d90a39057508d143d024779ee","target":"graph","created_at":"2026-05-18T03:42:23Z","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":"For maximal planar graphs of order $n\\geq 4$, we prove that a vertex--coloring containing no rainbow faces uses at most $\\lfloor\\frac{2n-1}{3}\\rfloor$ colors, and this is best possible. For maximal graph embedded on the projective plane, we obtain the analogous best bound $\\lfloor\\frac{2n+1}{3}\\rfloor$. The main ingredients in the proofs are classical homological tools. By considering graphs as topological spaces, we introduce the notion of a null coloring, and prove that for any graph $G$ a maximal null coloring $f$ is such that the quotient graph $G/f$ is a forest.","authors_text":"Amanda Montejano, Jorge L. Arocha","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-10-25T13:38:38Z","title":"Null and non--rainbow colorings of projective plane and sphere triangulations"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1210.6831","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:f8d4134c35a14c376ea2781bae838448d9e4c973a7df6eb6acf0786fc10be812","target":"record","created_at":"2026-05-18T03:42:23Z","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":"e996199800cc3010d49f61a5ee45c88d9a9f1df52b3093a4a0c924f70f88b79f","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-10-25T13:38:38Z","title_canon_sha256":"980284b031b81ec5da190e1797aa15ac7b6029c60848b56766d8a8670114de10"},"schema_version":"1.0","source":{"id":"1210.6831","kind":"arxiv","version":1}},"canonical_sha256":"77a8d7bf471868b3ed48ab52dfe856ea7ad346841f251427af76c3fb5adc95b6","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"77a8d7bf471868b3ed48ab52dfe856ea7ad346841f251427af76c3fb5adc95b6","first_computed_at":"2026-05-18T03:42:23.010003Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:42:23.010003Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"psPH/4NDIqgIcbR2pUlZP4Jw38he+Z3nRY0ZFonfSdIFKCVYZLrvqvhcpvr+hisK1vfbQ33ejGMQiml9KI/fAg==","signature_status":"signed_v1","signed_at":"2026-05-18T03:42:23.010588Z","signed_message":"canonical_sha256_bytes"},"source_id":"1210.6831","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:f8d4134c35a14c376ea2781bae838448d9e4c973a7df6eb6acf0786fc10be812","sha256:ab96ec671d18386753ec4657e5920c8c0bd9cd9d90a39057508d143d024779ee"],"state_sha256":"8851219734f8bc29a3cc898121434908121c223ccabc8a2e39a61747d60b6335"}