{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:ZZCYHBERRFH5PNEGTIBZX7EVYO","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":"ea4b7158c60273f72b17bfd0c4b8c5641b1cb1973849488caeca57df429f6660","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-09-25T13:50:44Z","title_canon_sha256":"fc12106c9457a02447a21c9cbb92c63156411880dad29e1f757a5e1f7ed1bd67"},"schema_version":"1.0","source":{"id":"1509.07716","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1509.07716","created_at":"2026-05-18T00:09:00Z"},{"alias_kind":"arxiv_version","alias_value":"1509.07716v2","created_at":"2026-05-18T00:09:00Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1509.07716","created_at":"2026-05-18T00:09:00Z"},{"alias_kind":"pith_short_12","alias_value":"ZZCYHBERRFH5","created_at":"2026-05-18T12:29:52Z"},{"alias_kind":"pith_short_16","alias_value":"ZZCYHBERRFH5PNEG","created_at":"2026-05-18T12:29:52Z"},{"alias_kind":"pith_short_8","alias_value":"ZZCYHBER","created_at":"2026-05-18T12:29:52Z"}],"graph_snapshots":[{"event_id":"sha256:1eeec11053e376deae7874ae740f4e6dce9accc8eacec0f5be3a53dd7f6d27e9","target":"graph","created_at":"2026-05-18T00:09:00Z","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":"We show that every $4$-chromatic graph on $n$ vertices, with no two vertex-disjoint odd cycles, has an odd cycle of length at most $\\tfrac12\\,(1+\\sqrt{8n-7})$. Let $G$ be a non-bipartite quadrangulation of the projective plane on $n$ vertices. Our result immediately implies that $G$ has edge-width at most $\\tfrac12\\,(1+\\sqrt{8n-7})$, which is sharp for infinitely many values of $n$. We also show that $G$ has face-width (equivalently, contains an odd cycle transversal of cardinality) at most $\\tfrac14(1+\\sqrt{16 n-15})$, which is a constant away from the optimal; we prove a lower bound of $\\sqr","authors_text":"Louis Esperet, Mat\\v{e}j Stehl\\'ik","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-09-25T13:50:44Z","title":"The width of quadrangulations of the projective plane"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1509.07716","kind":"arxiv","version":2},"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":"sha256:075c95d415970ecc5f8a9b4ced44e0e6cb2f1f9cf8ceae1b262488c2c5247ffa","target":"record","created_at":"2026-05-18T00:09:00Z","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":"ea4b7158c60273f72b17bfd0c4b8c5641b1cb1973849488caeca57df429f6660","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-09-25T13:50:44Z","title_canon_sha256":"fc12106c9457a02447a21c9cbb92c63156411880dad29e1f757a5e1f7ed1bd67"},"schema_version":"1.0","source":{"id":"1509.07716","kind":"arxiv","version":2}},"canonical_sha256":"ce45838491894fd7b4869a039bfc95c3a0c239e0ee46f9dc117dc96430b45689","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"ce45838491894fd7b4869a039bfc95c3a0c239e0ee46f9dc117dc96430b45689","first_computed_at":"2026-05-18T00:09:00.863397Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:09:00.863397Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"UZks9ifYXVq3nk+AdXETFbi4iTZ7oLB0pTgw0wGi2a2ZNDNcweKDmE7/Nbig9SSw1Hxzdnfb7/Oc+I9yF8+GDg==","signature_status":"signed_v1","signed_at":"2026-05-18T00:09:00.863972Z","signed_message":"canonical_sha256_bytes"},"source_id":"1509.07716","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:075c95d415970ecc5f8a9b4ced44e0e6cb2f1f9cf8ceae1b262488c2c5247ffa","sha256:1eeec11053e376deae7874ae740f4e6dce9accc8eacec0f5be3a53dd7f6d27e9"],"state_sha256":"25d6a0921d7d1730ed510b043e977af378e6c7eac837aae1cebef1ef587d15fb"}