{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:NHXKSBWYOVB6XMQT5AMUJRDMMN","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":"0a242e3291f8aeeeff77ffdf4d93b614f5ae2c4ae2cfa086f5711de2c4aa1817","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-02-15T18:22:19Z","title_canon_sha256":"9e72a5360bf56f374a0a5250ad90d213886712d13c32913398d64912a53dc084"},"schema_version":"1.0","source":{"id":"1102.3147","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1102.3147","created_at":"2026-05-18T04:28:38Z"},{"alias_kind":"arxiv_version","alias_value":"1102.3147v1","created_at":"2026-05-18T04:28:38Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1102.3147","created_at":"2026-05-18T04:28:38Z"},{"alias_kind":"pith_short_12","alias_value":"NHXKSBWYOVB6","created_at":"2026-05-18T12:26:37Z"},{"alias_kind":"pith_short_16","alias_value":"NHXKSBWYOVB6XMQT","created_at":"2026-05-18T12:26:37Z"},{"alias_kind":"pith_short_8","alias_value":"NHXKSBWY","created_at":"2026-05-18T12:26:37Z"}],"graph_snapshots":[{"event_id":"sha256:1ddf3600bef425f311597f9713585af7670527e74d2899afe0219c0b9bc850e7","target":"graph","created_at":"2026-05-18T04:28:38Z","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":"Long paths and cycles in sparse random graphs and digraphs were studied intensively in the 1980's. It was finally shown by Frieze in 1986 that the random graph $\\cG(n,p)$ with $p=c/n$ has a cycle on at all but at most $(1+\\epsilon)ce^{-c}n$ vertices with high probability, where $\\epsilon=\\epsilon(c)\\to 0$ as $c\\to\\infty$. This estimate on the number of uncovered vertices is essentially tight due to vertices of degree 1. However, for the random digraph $\\cD(n,p)$ no tight result was known and the best estimate was a factor of $c/2$ away from the corresponding lower bound. In this work we close ","authors_text":"Benny Sudakov, Eyal Lubetzky, Michael Krivelevich","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-02-15T18:22:19Z","title":"Longest cycles in sparse random digraphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1102.3147","kind":"arxiv","version":1},"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":"sha256:51c906ced0c10081708e3fe06ef2e5b646b2bf70bd4b2cc0fc59c1d7b3342129","target":"record","created_at":"2026-05-18T04:28:38Z","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":"0a242e3291f8aeeeff77ffdf4d93b614f5ae2c4ae2cfa086f5711de2c4aa1817","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-02-15T18:22:19Z","title_canon_sha256":"9e72a5360bf56f374a0a5250ad90d213886712d13c32913398d64912a53dc084"},"schema_version":"1.0","source":{"id":"1102.3147","kind":"arxiv","version":1}},"canonical_sha256":"69eea906d87543ebb213e81944c46c637c11c199646558e55259535787f68d7f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"69eea906d87543ebb213e81944c46c637c11c199646558e55259535787f68d7f","first_computed_at":"2026-05-18T04:28:38.326727Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:28:38.326727Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"+WWYwtB8u9xLlsHC00pxY8/E62EmXBouRtKVN7Scgycv++tZSndUCCtCkijkuchxLtk5buH/3ucydAUNWPogAw==","signature_status":"signed_v1","signed_at":"2026-05-18T04:28:38.327113Z","signed_message":"canonical_sha256_bytes"},"source_id":"1102.3147","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:51c906ced0c10081708e3fe06ef2e5b646b2bf70bd4b2cc0fc59c1d7b3342129","sha256:1ddf3600bef425f311597f9713585af7670527e74d2899afe0219c0b9bc850e7"],"state_sha256":"5c61766b934dc56edbb46cbc19d3a186c3af57e5cb6dd28029ab4965739a78c5"}