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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"},"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":"2604.01068","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-04-01T16:05:34Z","cross_cats_sorted":[],"title_canon_sha256":"b2d18299a2da072c0eef52a9368653d53f0eabcba0991a8a09b7dd804915d0f1","abstract_canon_sha256":"8934bfb30b26b0090bc235fce75b35f95856f091d3afefa6c81153b2111f3fef"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-08T01:19:12.237731Z","signature_b64":"cuG6NfnlguxQOJ5RYgCOKpoKPgUjsvSbnS6kFyrFb49+wBjCrPyXG/wGRvonM/mv19gVCRwOykL3HO3IKiFOBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6052d5da60cd4856531d2d4b9d0dcc64c9e96965b9ef47c27912359d2ffbe60b","last_reissued_at":"2026-07-08T01:19:12.237164Z","signature_status":"signed_v1","first_computed_at":"2026-07-08T01:19:12.237164Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Extensions of Erd\\H{o}s's 1962 theorem on non-Hamiltonian graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Bo Ning, Tao Wang, Xu Liu","submitted_at":"2026-04-01T16:05:34Z","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"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2604.01068","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2604.01068/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2604.01068","created_at":"2026-07-08T01:19:12.237229+00:00"},{"alias_kind":"arxiv_version","alias_value":"2604.01068v2","created_at":"2026-07-08T01:19:12.237229+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2604.01068","created_at":"2026-07-08T01:19:12.237229+00:00"},{"alias_kind":"pith_short_12","alias_value":"MBJNLWTAZVEF","created_at":"2026-07-08T01:19:12.237229+00:00"},{"alias_kind":"pith_short_16","alias_value":"MBJNLWTAZVEFMUY5","created_at":"2026-07-08T01:19:12.237229+00:00"},{"alias_kind":"pith_short_8","alias_value":"MBJNLWTA","created_at":"2026-07-08T01:19:12.237229+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"2607.07064","citing_title":"Sharp spectral Moon--Moser-type theorems in the linear range via feasible graph parameters","ref_index":10,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/MBJNLWTAZVEFMUY5FVFZ2DOMMT","json":"https://pith.science/pith/MBJNLWTAZVEFMUY5FVFZ2DOMMT.json","graph_json":"https://pith.science/api/pith-number/MBJNLWTAZVEFMUY5FVFZ2DOMMT/graph.json","events_json":"https://pith.science/api/pith-number/MBJNLWTAZVEFMUY5FVFZ2DOMMT/events.json","paper":"https://pith.science/paper/MBJNLWTA"},"agent_actions":{"view_html":"https://pith.science/pith/MBJNLWTAZVEFMUY5FVFZ2DOMMT","download_json":"https://pith.science/pith/MBJNLWTAZVEFMUY5FVFZ2DOMMT.json","view_paper":"https://pith.science/paper/MBJNLWTA","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2604.01068&json=true","fetch_graph":"https://pith.science/api/pith-number/MBJNLWTAZVEFMUY5FVFZ2DOMMT/graph.json","fetch_events":"https://pith.science/api/pith-number/MBJNLWTAZVEFMUY5FVFZ2DOMMT/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/MBJNLWTAZVEFMUY5FVFZ2DOMMT/action/timestamp_anchor","attest_storage":"https://pith.science/pith/MBJNLWTAZVEFMUY5FVFZ2DOMMT/action/storage_attestation","attest_author":"https://pith.science/pith/MBJNLWTAZVEFMUY5FVFZ2DOMMT/action/author_attestation","sign_citation":"https://pith.science/pith/MBJNLWTAZVEFMUY5FVFZ2DOMMT/action/citation_signature","submit_replication":"https://pith.science/pith/MBJNLWTAZVEFMUY5FVFZ2DOMMT/action/replication_record"}},"created_at":"2026-07-08T01:19:12.237229+00:00","updated_at":"2026-07-08T01:19:12.237229+00:00"}