{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2013:43FHU6G4SAHWMQ33K2LJ3K4BIG","short_pith_number":"pith:43FHU6G4","schema_version":"1.0","canonical_sha256":"e6ca7a78dc900f66437b56969dab8141ade6bf76f2465c6be87466d0296bb876","source":{"kind":"arxiv","id":"1309.3665","version":2},"attestation_state":"computed","paper":{"title":"Shellable drawings and the cylindrical crossing number of $K_n$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.CG"],"primary_cat":"math.CO","authors_text":"Bernardo M. \\'Abrego, Gelasio Salazar, Oswin Aichholzer, Pedro Ramos, Silvia Fern\\'andez-Merchant","submitted_at":"2013-09-14T13:24:20Z","abstract_excerpt":"The Harary-Hill Conjecture States that the number of crossings in any drawing of the complete graph $ K_n $ in the plane is at least $Z(n):=\\frac{1}{4}\\left\\lfloor \\frac{n}{2}\\right\\rfloor \\left\\lfloor\\frac{n-1}{2}\\right\\rfloor \\left\\lfloor \\frac{n-2}{2}\\right\\rfloor\\left\\lfloor \\frac{n-3}{2}\\right\\rfloor$. In this paper, we settle the Harary-Hill conjecture for {\\em shellable drawings}. We say that a drawing $D$ of $ K_n $ is {\\em $ s $-shellable} if there exist a subset $ S = \\{v_1,v_2,\\ldots,v_ s\\}$ of the vertices and a region $R$ of $D$ with the following property: For all $1 \\leq i < j \\"},"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":"1309.3665","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-09-14T13:24:20Z","cross_cats_sorted":["cs.CG"],"title_canon_sha256":"66ee686b7323d4afc633589af1bc46f374615b7c6a3fb3acebb5a8cfa3aa044b","abstract_canon_sha256":"d6d00104517e6f9ec845f0894fe5029bc48513c75e047bfe1786b34057cde182"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:10:45.519858Z","signature_b64":"YSbC6+wQGdbGzXOr3IdYsMt/tGeWUcj+O7KZhlSnqSO93M+4nyG6cyoWHXnkYVAR1lyxSuC381v6ZL7Mk4umAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e6ca7a78dc900f66437b56969dab8141ade6bf76f2465c6be87466d0296bb876","last_reissued_at":"2026-05-18T03:10:45.519406Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:10:45.519406Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Shellable drawings and the cylindrical crossing number of $K_n$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.CG"],"primary_cat":"math.CO","authors_text":"Bernardo M. \\'Abrego, Gelasio Salazar, Oswin Aichholzer, Pedro Ramos, Silvia Fern\\'andez-Merchant","submitted_at":"2013-09-14T13:24:20Z","abstract_excerpt":"The Harary-Hill Conjecture States that the number of crossings in any drawing of the complete graph $ K_n $ in the plane is at least $Z(n):=\\frac{1}{4}\\left\\lfloor \\frac{n}{2}\\right\\rfloor \\left\\lfloor\\frac{n-1}{2}\\right\\rfloor \\left\\lfloor \\frac{n-2}{2}\\right\\rfloor\\left\\lfloor \\frac{n-3}{2}\\right\\rfloor$. In this paper, we settle the Harary-Hill conjecture for {\\em shellable drawings}. We say that a drawing $D$ of $ K_n $ is {\\em $ s $-shellable} if there exist a subset $ S = \\{v_1,v_2,\\ldots,v_ s\\}$ of the vertices and a region $R$ of $D$ with the following property: For all $1 \\leq i < j \\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1309.3665","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":"1309.3665","created_at":"2026-05-18T03:10:45.519464+00:00"},{"alias_kind":"arxiv_version","alias_value":"1309.3665v2","created_at":"2026-05-18T03:10:45.519464+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1309.3665","created_at":"2026-05-18T03:10:45.519464+00:00"},{"alias_kind":"pith_short_12","alias_value":"43FHU6G4SAHW","created_at":"2026-05-18T12:27:32.513160+00:00"},{"alias_kind":"pith_short_16","alias_value":"43FHU6G4SAHWMQ33","created_at":"2026-05-18T12:27:32.513160+00:00"},{"alias_kind":"pith_short_8","alias_value":"43FHU6G4","created_at":"2026-05-18T12:27:32.513160+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/43FHU6G4SAHWMQ33K2LJ3K4BIG","json":"https://pith.science/pith/43FHU6G4SAHWMQ33K2LJ3K4BIG.json","graph_json":"https://pith.science/api/pith-number/43FHU6G4SAHWMQ33K2LJ3K4BIG/graph.json","events_json":"https://pith.science/api/pith-number/43FHU6G4SAHWMQ33K2LJ3K4BIG/events.json","paper":"https://pith.science/paper/43FHU6G4"},"agent_actions":{"view_html":"https://pith.science/pith/43FHU6G4SAHWMQ33K2LJ3K4BIG","download_json":"https://pith.science/pith/43FHU6G4SAHWMQ33K2LJ3K4BIG.json","view_paper":"https://pith.science/paper/43FHU6G4","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1309.3665&json=true","fetch_graph":"https://pith.science/api/pith-number/43FHU6G4SAHWMQ33K2LJ3K4BIG/graph.json","fetch_events":"https://pith.science/api/pith-number/43FHU6G4SAHWMQ33K2LJ3K4BIG/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/43FHU6G4SAHWMQ33K2LJ3K4BIG/action/timestamp_anchor","attest_storage":"https://pith.science/pith/43FHU6G4SAHWMQ33K2LJ3K4BIG/action/storage_attestation","attest_author":"https://pith.science/pith/43FHU6G4SAHWMQ33K2LJ3K4BIG/action/author_attestation","sign_citation":"https://pith.science/pith/43FHU6G4SAHWMQ33K2LJ3K4BIG/action/citation_signature","submit_replication":"https://pith.science/pith/43FHU6G4SAHWMQ33K2LJ3K4BIG/action/replication_record"}},"created_at":"2026-05-18T03:10:45.519464+00:00","updated_at":"2026-05-18T03:10:45.519464+00:00"}