{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2020:5GDBVIO5WQIPOPAAB4QEZDBA5C","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":"fa7b4903606adf34e45567d0be9e960ddb7c73fcac1ee78d9ae72170ef51a1ea","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DS","submitted_at":"2020-11-06T22:32:33Z","title_canon_sha256":"48c7633bfc9a05793d342b84118d27b6653bc8754c24ef443c684466ecdedeeb"},"schema_version":"1.0","source":{"id":"2011.03619","kind":"arxiv","version":5}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2011.03619","created_at":"2026-07-05T08:07:06Z"},{"alias_kind":"arxiv_version","alias_value":"2011.03619v5","created_at":"2026-07-05T08:07:06Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2011.03619","created_at":"2026-07-05T08:07:06Z"},{"alias_kind":"pith_short_12","alias_value":"5GDBVIO5WQIP","created_at":"2026-07-05T08:07:06Z"},{"alias_kind":"pith_short_16","alias_value":"5GDBVIO5WQIPOPAA","created_at":"2026-07-05T08:07:06Z"},{"alias_kind":"pith_short_8","alias_value":"5GDBVIO5","created_at":"2026-07-05T08:07:06Z"}],"graph_snapshots":[{"event_id":"sha256:45c7240dc9c8905d463820afda64e4a6ec659e47e55441f30fe2bc3355cd8438","target":"graph","created_at":"2026-07-05T08:07:06Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2011.03619/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"In 1952, Dirac proved the following theorem about long cycles in graphs with large minimum vertex degrees: Every $n$-vertex $2$-connected graph $G$ with minimum vertex degree $\\delta\\geq 2$ contains a cycle with at least $\\min\\{2\\delta,n\\}$ vertices. In particular, if $\\delta\\geq n/2$, then $G$ is Hamiltonian. The proof of Dirac's theorem is constructive, and it yields an algorithm computing the corresponding cycle in polynomial time. The combinatorial bound of Dirac's theorem is tight in the following sense. There are 2-connected graphs that do not contain cycles of length more than $2\\delta+","authors_text":"Danil Sagunov, Fedor V. Fomin, Kirill Simonov, Petr A. Golovach","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DS","submitted_at":"2020-11-06T22:32:33Z","title":"Algorithmic Extensions of Dirac's Theorem"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2011.03619","kind":"arxiv","version":5},"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":"sha256:5be40940fa1f090a5752b0b0829a6171c930c2e96a69aba2c6b174169e97557d","target":"record","created_at":"2026-07-05T08:07:06Z","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":"fa7b4903606adf34e45567d0be9e960ddb7c73fcac1ee78d9ae72170ef51a1ea","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DS","submitted_at":"2020-11-06T22:32:33Z","title_canon_sha256":"48c7633bfc9a05793d342b84118d27b6653bc8754c24ef443c684466ecdedeeb"},"schema_version":"1.0","source":{"id":"2011.03619","kind":"arxiv","version":5}},"canonical_sha256":"e9861aa1ddb410f73c000f204c8c20e8b99563de1be3bcff5ba3163c4ebb2d10","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"e9861aa1ddb410f73c000f204c8c20e8b99563de1be3bcff5ba3163c4ebb2d10","first_computed_at":"2026-07-05T08:07:06.981596Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T08:07:06.981596Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"jcB+MVWPMMujasWCqrsWi9MzIblKJoka7w3UXKdBTUoP0ZhqLy7naZ4CnRpmZ0tA5tySGxp10MRuaiiWhrbaAw==","signature_status":"signed_v1","signed_at":"2026-07-05T08:07:06.981970Z","signed_message":"canonical_sha256_bytes"},"source_id":"2011.03619","source_kind":"arxiv","source_version":5}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:5be40940fa1f090a5752b0b0829a6171c930c2e96a69aba2c6b174169e97557d","sha256:45c7240dc9c8905d463820afda64e4a6ec659e47e55441f30fe2bc3355cd8438"],"state_sha256":"e16e257b13c313e0ea41a500dbdd909f96ba36ca325353311b3bae60818625d3"}