{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:MBJNLWTAZVEFMUY5FVFZ2DOMMT","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":"8934bfb30b26b0090bc235fce75b35f95856f091d3afefa6c81153b2111f3fef","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-04-01T16:05:34Z","title_canon_sha256":"b2d18299a2da072c0eef52a9368653d53f0eabcba0991a8a09b7dd804915d0f1"},"schema_version":"1.0","source":{"id":"2604.01068","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2604.01068","created_at":"2026-07-08T01:19:12Z"},{"alias_kind":"arxiv_version","alias_value":"2604.01068v2","created_at":"2026-07-08T01:19:12Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2604.01068","created_at":"2026-07-08T01:19:12Z"},{"alias_kind":"pith_short_12","alias_value":"MBJNLWTAZVEF","created_at":"2026-07-08T01:19:12Z"},{"alias_kind":"pith_short_16","alias_value":"MBJNLWTAZVEFMUY5","created_at":"2026-07-08T01:19:12Z"},{"alias_kind":"pith_short_8","alias_value":"MBJNLWTA","created_at":"2026-07-08T01:19:12Z"}],"graph_snapshots":[{"event_id":"sha256:700c92f1d9c0cbde12e6dce1878e8f4fe11643a0fb3ca80042ab6dd8a220bae4","target":"graph","created_at":"2026-07-08T01:19:12Z","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/2604.01068/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"For a positive integer $k$, a graph property $\\mathcal{H}$, and a graph parameter $\\mathcal{P}$, let $\\operatorname{ex}_{\\mathcal{P}}(n, \\mathcal{H}; \\delta \\geq k)$ denote the maximum value of $\\mathcal{P}$ over all $n$-vertex graphs with minimum degree at least $k$ that do not possess the property $\\mathcal{H}$. The corresponding extremal families are denoted by $\\operatorname{EX}_{\\mathcal{P}}(n, \\mathcal{H}; \\delta \\geq k)$. For two disjoint graphs $H_1$ and $H_2$, let $H_1 \\cup H_2$ denote their disjoint union, and let $H_1 \\vee H_2$ denote their join.\n  In 1962, Erd\\H{o}s established a c","authors_text":"Bo Ning, Tao Wang, Xu Liu","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-04-01T16:05:34Z","title":"Extensions of Erd\\H{o}s's 1962 theorem on non-Hamiltonian graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2604.01068","kind":"arxiv","version":2},"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":"sha256:15eb84a19c53c73d4620cb3953810dc6dfdbeb93340c6845dae1fdd9ee33a65b","target":"record","created_at":"2026-07-08T01:19:12Z","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":"8934bfb30b26b0090bc235fce75b35f95856f091d3afefa6c81153b2111f3fef","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-04-01T16:05:34Z","title_canon_sha256":"b2d18299a2da072c0eef52a9368653d53f0eabcba0991a8a09b7dd804915d0f1"},"schema_version":"1.0","source":{"id":"2604.01068","kind":"arxiv","version":2}},"canonical_sha256":"6052d5da60cd4856531d2d4b9d0dcc64c9e96965b9ef47c27912359d2ffbe60b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"6052d5da60cd4856531d2d4b9d0dcc64c9e96965b9ef47c27912359d2ffbe60b","first_computed_at":"2026-07-08T01:19:12.237164Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-08T01:19:12.237164Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"cuG6NfnlguxQOJ5RYgCOKpoKPgUjsvSbnS6kFyrFb49+wBjCrPyXG/wGRvonM/mv19gVCRwOykL3HO3IKiFOBA==","signature_status":"signed_v1","signed_at":"2026-07-08T01:19:12.237731Z","signed_message":"canonical_sha256_bytes"},"source_id":"2604.01068","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:15eb84a19c53c73d4620cb3953810dc6dfdbeb93340c6845dae1fdd9ee33a65b","sha256:700c92f1d9c0cbde12e6dce1878e8f4fe11643a0fb3ca80042ab6dd8a220bae4"],"state_sha256":"fe4f1de08fd6f3e1f913ab39ad726731671d7afa4618071ad48012c0de6667c2"}