{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2005:ZPIJAL6PRPAXTIDD36IACKBGDO","short_pith_number":"pith:ZPIJAL6P","schema_version":"1.0","canonical_sha256":"cbd0902fcf8bc179a063df900128261b885b481225f12a16785285b746338727","source":{"kind":"arxiv","id":"math/0512077","version":4},"attestation_state":"computed","paper":{"title":"The neighborhood complex of a random graph","license":"","headline":"","cross_cats":["math.AT"],"primary_cat":"math.CO","authors_text":"Matthew Kahle","submitted_at":"2005-12-04T07:16:35Z","abstract_excerpt":"For a graph G, the neighborhood complex N[G] is the simplicial complex having all subsets of vertices with a common neighbor as its faces. It is a well known result of Lovasz that if N[G] is k-connected, then the chromatic number of G is at least k + 3.\n  We prove that the connectivity of the neighborhood complex of a random graph is tightly concentrated, almost always between 1/2 and 2/3 of the expected clique number. We also show that the number of dimensions of nontrivial homology is almost always small, O(log d), compared to the expected dimension d of the complex itself."},"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":"math/0512077","kind":"arxiv","version":4},"metadata":{"license":"","primary_cat":"math.CO","submitted_at":"2005-12-04T07:16:35Z","cross_cats_sorted":["math.AT"],"title_canon_sha256":"60e15b5472712d4669ec9c1e963f83017fe3749e8c7467a664fb954ac88d3cc7","abstract_canon_sha256":"58ea5237ccfccf2d6234de4cdc00094591a568316c7b2bbe5ff1366befac6bdb"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:40:33.041079Z","signature_b64":"IageCuaf+v10mX0pLzmKhl7Wo5pH24zcOuWA7LPHPA0tVh3TSOQA2x1hJzkem6O6xeJSM9pPhd9cwuh4uh4GDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"cbd0902fcf8bc179a063df900128261b885b481225f12a16785285b746338727","last_reissued_at":"2026-05-18T04:40:33.040217Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:40:33.040217Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The neighborhood complex of a random graph","license":"","headline":"","cross_cats":["math.AT"],"primary_cat":"math.CO","authors_text":"Matthew Kahle","submitted_at":"2005-12-04T07:16:35Z","abstract_excerpt":"For a graph G, the neighborhood complex N[G] is the simplicial complex having all subsets of vertices with a common neighbor as its faces. It is a well known result of Lovasz that if N[G] is k-connected, then the chromatic number of G is at least k + 3.\n  We prove that the connectivity of the neighborhood complex of a random graph is tightly concentrated, almost always between 1/2 and 2/3 of the expected clique number. We also show that the number of dimensions of nontrivial homology is almost always small, O(log d), compared to the expected dimension d of the complex itself."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0512077","kind":"arxiv","version":4},"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":"math/0512077","created_at":"2026-05-18T04:40:33.040335+00:00"},{"alias_kind":"arxiv_version","alias_value":"math/0512077v4","created_at":"2026-05-18T04:40:33.040335+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0512077","created_at":"2026-05-18T04:40:33.040335+00:00"},{"alias_kind":"pith_short_12","alias_value":"ZPIJAL6PRPAX","created_at":"2026-05-18T12:25:53.939244+00:00"},{"alias_kind":"pith_short_16","alias_value":"ZPIJAL6PRPAXTIDD","created_at":"2026-05-18T12:25:53.939244+00:00"},{"alias_kind":"pith_short_8","alias_value":"ZPIJAL6P","created_at":"2026-05-18T12:25:53.939244+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/ZPIJAL6PRPAXTIDD36IACKBGDO","json":"https://pith.science/pith/ZPIJAL6PRPAXTIDD36IACKBGDO.json","graph_json":"https://pith.science/api/pith-number/ZPIJAL6PRPAXTIDD36IACKBGDO/graph.json","events_json":"https://pith.science/api/pith-number/ZPIJAL6PRPAXTIDD36IACKBGDO/events.json","paper":"https://pith.science/paper/ZPIJAL6P"},"agent_actions":{"view_html":"https://pith.science/pith/ZPIJAL6PRPAXTIDD36IACKBGDO","download_json":"https://pith.science/pith/ZPIJAL6PRPAXTIDD36IACKBGDO.json","view_paper":"https://pith.science/paper/ZPIJAL6P","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=math/0512077&json=true","fetch_graph":"https://pith.science/api/pith-number/ZPIJAL6PRPAXTIDD36IACKBGDO/graph.json","fetch_events":"https://pith.science/api/pith-number/ZPIJAL6PRPAXTIDD36IACKBGDO/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/ZPIJAL6PRPAXTIDD36IACKBGDO/action/timestamp_anchor","attest_storage":"https://pith.science/pith/ZPIJAL6PRPAXTIDD36IACKBGDO/action/storage_attestation","attest_author":"https://pith.science/pith/ZPIJAL6PRPAXTIDD36IACKBGDO/action/author_attestation","sign_citation":"https://pith.science/pith/ZPIJAL6PRPAXTIDD36IACKBGDO/action/citation_signature","submit_replication":"https://pith.science/pith/ZPIJAL6PRPAXTIDD36IACKBGDO/action/replication_record"}},"created_at":"2026-05-18T04:40:33.040335+00:00","updated_at":"2026-05-18T04:40:33.040335+00:00"}