{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:Y2C6DFYBWQAXRHSLBAMNRUB56F","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":"2a9444418efe1900b9aeb83249e822fe5a59af56a6d78aac2715bcf198f7c76b","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-07-02T09:15:38Z","title_canon_sha256":"2e2b37d5f5f340e798c444f4ff8ff810319b68a56dad86d4c6f1a22b67b79469"},"schema_version":"1.0","source":{"id":"1207.0312","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1207.0312","created_at":"2026-05-18T03:24:54Z"},{"alias_kind":"arxiv_version","alias_value":"1207.0312v3","created_at":"2026-05-18T03:24:54Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1207.0312","created_at":"2026-05-18T03:24:54Z"},{"alias_kind":"pith_short_12","alias_value":"Y2C6DFYBWQAX","created_at":"2026-05-18T12:27:27Z"},{"alias_kind":"pith_short_16","alias_value":"Y2C6DFYBWQAXRHSL","created_at":"2026-05-18T12:27:27Z"},{"alias_kind":"pith_short_8","alias_value":"Y2C6DFYB","created_at":"2026-05-18T12:27:27Z"}],"graph_snapshots":[{"event_id":"sha256:86235a48aeb41fbaaaa0f286d9a62b15292a4d566fe1eb3220fa0db8bb15289e","target":"graph","created_at":"2026-05-18T03:24:54Z","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":"For a given finite graph $G$ of minimum degree at least $k$, let $G_{p}$ be a random subgraph of $G$ obtained by taking each edge independently with probability $p$. We prove that (i) if $p \\ge \\omega/k$ for a function $\\omega=\\omega(k)$ that tends to infinity as $k$ does, then $G_p$ asymptotically almost surely contains a cycle (and thus a path) of length at least $(1-o(1))k$, and (ii) if $p \\ge (1+o(1))\\ln k/k$, then $G_p$ asymptotically almost surely contains a path of length at least $k$. Our theorems extend classical results on paths and cycles in the binomial random graph, obtained by ta","authors_text":"Benny Sudakov, Choongbum Lee, Michael Krivelevich","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-07-02T09:15:38Z","title":"Long paths and cycles in random subgraphs of graphs with large minimum degree"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1207.0312","kind":"arxiv","version":3},"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":"sha256:3bd8bb9f7dde8091d8ddc5bfc42e89ef476a028019548434343fd54ec0fb62c6","target":"record","created_at":"2026-05-18T03:24:54Z","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":"2a9444418efe1900b9aeb83249e822fe5a59af56a6d78aac2715bcf198f7c76b","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-07-02T09:15:38Z","title_canon_sha256":"2e2b37d5f5f340e798c444f4ff8ff810319b68a56dad86d4c6f1a22b67b79469"},"schema_version":"1.0","source":{"id":"1207.0312","kind":"arxiv","version":3}},"canonical_sha256":"c685e19701b401789e4b0818d8d03df1415f0098f9cf7eae298469d894364b91","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"c685e19701b401789e4b0818d8d03df1415f0098f9cf7eae298469d894364b91","first_computed_at":"2026-05-18T03:24:54.246153Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:24:54.246153Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"UuwEP0HtDNjJSB4wcCL7wyOUO0iASrYCdhmovkreTMXRe3oH0mYvFSH5Rr1fd0NKJiQwIqZNkNWHz7rrDebtDg==","signature_status":"signed_v1","signed_at":"2026-05-18T03:24:54.246833Z","signed_message":"canonical_sha256_bytes"},"source_id":"1207.0312","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:3bd8bb9f7dde8091d8ddc5bfc42e89ef476a028019548434343fd54ec0fb62c6","sha256:86235a48aeb41fbaaaa0f286d9a62b15292a4d566fe1eb3220fa0db8bb15289e"],"state_sha256":"39e691c0de3a45135e522221b8032bd804302357a8a197bb9bf19aea7ec79ca6"}