{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2015:ZZCYHBERRFH5PNEGTIBZX7EVYO","short_pith_number":"pith:ZZCYHBER","schema_version":"1.0","canonical_sha256":"ce45838491894fd7b4869a039bfc95c3a0c239e0ee46f9dc117dc96430b45689","source":{"kind":"arxiv","id":"1509.07716","version":2},"attestation_state":"computed","paper":{"title":"The width of quadrangulations of the projective plane","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Louis Esperet, Mat\\v{e}j Stehl\\'ik","submitted_at":"2015-09-25T13:50:44Z","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"},"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":"1509.07716","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-09-25T13:50:44Z","cross_cats_sorted":[],"title_canon_sha256":"fc12106c9457a02447a21c9cbb92c63156411880dad29e1f757a5e1f7ed1bd67","abstract_canon_sha256":"ea4b7158c60273f72b17bfd0c4b8c5641b1cb1973849488caeca57df429f6660"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:09:00.863972Z","signature_b64":"UZks9ifYXVq3nk+AdXETFbi4iTZ7oLB0pTgw0wGi2a2ZNDNcweKDmE7/Nbig9SSw1Hxzdnfb7/Oc+I9yF8+GDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ce45838491894fd7b4869a039bfc95c3a0c239e0ee46f9dc117dc96430b45689","last_reissued_at":"2026-05-18T00:09:00.863397Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:09:00.863397Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The width of quadrangulations of the projective plane","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Louis Esperet, Mat\\v{e}j Stehl\\'ik","submitted_at":"2015-09-25T13:50:44Z","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"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1509.07716","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":""},"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":"1509.07716","created_at":"2026-05-18T00:09:00.863469+00:00"},{"alias_kind":"arxiv_version","alias_value":"1509.07716v2","created_at":"2026-05-18T00:09:00.863469+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1509.07716","created_at":"2026-05-18T00:09:00.863469+00:00"},{"alias_kind":"pith_short_12","alias_value":"ZZCYHBERRFH5","created_at":"2026-05-18T12:29:52.810259+00:00"},{"alias_kind":"pith_short_16","alias_value":"ZZCYHBERRFH5PNEG","created_at":"2026-05-18T12:29:52.810259+00:00"},{"alias_kind":"pith_short_8","alias_value":"ZZCYHBER","created_at":"2026-05-18T12:29:52.810259+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/ZZCYHBERRFH5PNEGTIBZX7EVYO","json":"https://pith.science/pith/ZZCYHBERRFH5PNEGTIBZX7EVYO.json","graph_json":"https://pith.science/api/pith-number/ZZCYHBERRFH5PNEGTIBZX7EVYO/graph.json","events_json":"https://pith.science/api/pith-number/ZZCYHBERRFH5PNEGTIBZX7EVYO/events.json","paper":"https://pith.science/paper/ZZCYHBER"},"agent_actions":{"view_html":"https://pith.science/pith/ZZCYHBERRFH5PNEGTIBZX7EVYO","download_json":"https://pith.science/pith/ZZCYHBERRFH5PNEGTIBZX7EVYO.json","view_paper":"https://pith.science/paper/ZZCYHBER","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1509.07716&json=true","fetch_graph":"https://pith.science/api/pith-number/ZZCYHBERRFH5PNEGTIBZX7EVYO/graph.json","fetch_events":"https://pith.science/api/pith-number/ZZCYHBERRFH5PNEGTIBZX7EVYO/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/ZZCYHBERRFH5PNEGTIBZX7EVYO/action/timestamp_anchor","attest_storage":"https://pith.science/pith/ZZCYHBERRFH5PNEGTIBZX7EVYO/action/storage_attestation","attest_author":"https://pith.science/pith/ZZCYHBERRFH5PNEGTIBZX7EVYO/action/author_attestation","sign_citation":"https://pith.science/pith/ZZCYHBERRFH5PNEGTIBZX7EVYO/action/citation_signature","submit_replication":"https://pith.science/pith/ZZCYHBERRFH5PNEGTIBZX7EVYO/action/replication_record"}},"created_at":"2026-05-18T00:09:00.863469+00:00","updated_at":"2026-05-18T00:09:00.863469+00:00"}