{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2010:LURBG3QQI2EPS6LYLOGH3GSHGG","short_pith_number":"pith:LURBG3QQ","schema_version":"1.0","canonical_sha256":"5d22136e104688f979785b8c7d9a47319ffa84cd310a631140bf306cd9e6666d","source":{"kind":"arxiv","id":"1001.0461","version":1},"attestation_state":"computed","paper":{"title":"Rank-width of Random Graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Choongbum Lee, Joonkyung Lee, Sang-il Oum","submitted_at":"2010-01-04T08:59:46Z","abstract_excerpt":"Rank-width of a graph G, denoted by rw(G), is a width parameter of graphs introduced by Oum and Seymour (2006).\n  We investigate the asymptotic behavior of rank-width of a random graph G(n,p). We show that, asymptotically almost surely, (i) if 0<p<1 is a constant, then rw(G(n,p)) = \\lceil n/3 \\rceil-O(1), (ii) if 1/n<< p <1/2, then rw(G(n,p))= \\lceil n/3\\rceil-o(n), (iii) if p = c/n and c > 1, then rw(G(n,p)) > r n for some r = r(c), and (iv) if p <= c/n and c<1, then rw(G(n,p)) <=2.\n  As a corollary, we deduce that G(n,p) has linear tree-width whenever p=c/n for each c>1, answering a question"},"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":"1001.0461","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2010-01-04T08:59:46Z","cross_cats_sorted":[],"title_canon_sha256":"3fed732465536e8cc1a281b6c992453461703e05afac32eca3ccfb18baac2586","abstract_canon_sha256":"31129f9c549d3813a2e0331b284ff1f3a5566ef76d27994dfee87d68cbc40a94"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:55:39.652025Z","signature_b64":"fjBZ+/YkScX/3MttJHzh7QHBe8NTECW/CawvrGcOFVjA9UJbsBLQJKqy9QX7GoHMKjUKyO2dAg0ucM9lxoL8Cw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5d22136e104688f979785b8c7d9a47319ffa84cd310a631140bf306cd9e6666d","last_reissued_at":"2026-05-18T02:55:39.651482Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:55:39.651482Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Rank-width of Random Graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Choongbum Lee, Joonkyung Lee, Sang-il Oum","submitted_at":"2010-01-04T08:59:46Z","abstract_excerpt":"Rank-width of a graph G, denoted by rw(G), is a width parameter of graphs introduced by Oum and Seymour (2006).\n  We investigate the asymptotic behavior of rank-width of a random graph G(n,p). We show that, asymptotically almost surely, (i) if 0<p<1 is a constant, then rw(G(n,p)) = \\lceil n/3 \\rceil-O(1), (ii) if 1/n<< p <1/2, then rw(G(n,p))= \\lceil n/3\\rceil-o(n), (iii) if p = c/n and c > 1, then rw(G(n,p)) > r n for some r = r(c), and (iv) if p <= c/n and c<1, then rw(G(n,p)) <=2.\n  As a corollary, we deduce that G(n,p) has linear tree-width whenever p=c/n for each c>1, answering a question"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1001.0461","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1001.0461","created_at":"2026-05-18T02:55:39.651566+00:00"},{"alias_kind":"arxiv_version","alias_value":"1001.0461v1","created_at":"2026-05-18T02:55:39.651566+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1001.0461","created_at":"2026-05-18T02:55:39.651566+00:00"},{"alias_kind":"pith_short_12","alias_value":"LURBG3QQI2EP","created_at":"2026-05-18T12:26:10.704358+00:00"},{"alias_kind":"pith_short_16","alias_value":"LURBG3QQI2EPS6LY","created_at":"2026-05-18T12:26:10.704358+00:00"},{"alias_kind":"pith_short_8","alias_value":"LURBG3QQ","created_at":"2026-05-18T12:26:10.704358+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"2603.08847","citing_title":"The Structure of Circle Graph States","ref_index":71,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/LURBG3QQI2EPS6LYLOGH3GSHGG","json":"https://pith.science/pith/LURBG3QQI2EPS6LYLOGH3GSHGG.json","graph_json":"https://pith.science/api/pith-number/LURBG3QQI2EPS6LYLOGH3GSHGG/graph.json","events_json":"https://pith.science/api/pith-number/LURBG3QQI2EPS6LYLOGH3GSHGG/events.json","paper":"https://pith.science/paper/LURBG3QQ"},"agent_actions":{"view_html":"https://pith.science/pith/LURBG3QQI2EPS6LYLOGH3GSHGG","download_json":"https://pith.science/pith/LURBG3QQI2EPS6LYLOGH3GSHGG.json","view_paper":"https://pith.science/paper/LURBG3QQ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1001.0461&json=true","fetch_graph":"https://pith.science/api/pith-number/LURBG3QQI2EPS6LYLOGH3GSHGG/graph.json","fetch_events":"https://pith.science/api/pith-number/LURBG3QQI2EPS6LYLOGH3GSHGG/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/LURBG3QQI2EPS6LYLOGH3GSHGG/action/timestamp_anchor","attest_storage":"https://pith.science/pith/LURBG3QQI2EPS6LYLOGH3GSHGG/action/storage_attestation","attest_author":"https://pith.science/pith/LURBG3QQI2EPS6LYLOGH3GSHGG/action/author_attestation","sign_citation":"https://pith.science/pith/LURBG3QQI2EPS6LYLOGH3GSHGG/action/citation_signature","submit_replication":"https://pith.science/pith/LURBG3QQI2EPS6LYLOGH3GSHGG/action/replication_record"}},"created_at":"2026-05-18T02:55:39.651566+00:00","updated_at":"2026-05-18T02:55:39.651566+00:00"}