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Dragani\\'{c}, Keevash and M\\\"{u}yesser recently proved that every regular Dirac graph has $\\Omega(2^n)$ cyclic subsets, resolving a problem of Erd\\H{o}s and Faudree.\n  We determine the sharp asymptotic lower bound throughout the linear range below Dirac's threshold. Let $G$ be an $n$-vertex $d$-regular graph with $d=\\Omega(n)$ and $d<n/2$, then $$\n  \\operatorname{Cyc}(G)\\ge (q-o(1))2^{n/q}, \\quad \\text{where } \\quad q=\\left\\lfloor \\frac{n}{d+1}\\right\\rfloor \\ge "},"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":"2607.06551","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-07-07T17:53:58Z","cross_cats_sorted":[],"title_canon_sha256":"22560ab113ea16fbe54859b68ea4c699c09185ab3ad9443d5f2c0846570950dc","abstract_canon_sha256":"0de88dbfb4facc0bda0010546cd03f3764ac1b40d81a201d4c7ac690bb9c8166"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-08T01:19:29.676092Z","signature_b64":"KLSacLLwba8rk7uJw3sjREdQna4lgXsM06Nyu1TzUgH9gPgcHEuYdv9qd1dF9OgTwbreGbIhKYI4NGk4TiEwDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a9df4cdf9cf0c6052e83991ecf9088a1eb2149175e8e7996baa2484ba7c14e57","last_reissued_at":"2026-07-08T01:19:29.675653Z","signature_status":"signed_v1","first_computed_at":"2026-07-08T01:19:29.675653Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Tight Staircase Bounds for Cyclic Subsets below Dirac's Threshold","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Hong Liu, Lanchao Wang, Mengyuan Niu, Zhifei Yan","submitted_at":"2026-07-07T17:53:58Z","abstract_excerpt":"Let $\\operatorname{Cyc}(G)$ denote the number of cyclic subsets in a graph $G$, which are subsets that induce a Hamiltonian subgraph. Dragani\\'{c}, Keevash and M\\\"{u}yesser recently proved that every regular Dirac graph has $\\Omega(2^n)$ cyclic subsets, resolving a problem of Erd\\H{o}s and Faudree.\n  We determine the sharp asymptotic lower bound throughout the linear range below Dirac's threshold. Let $G$ be an $n$-vertex $d$-regular graph with $d=\\Omega(n)$ and $d<n/2$, then $$\n  \\operatorname{Cyc}(G)\\ge (q-o(1))2^{n/q}, \\quad \\text{where } \\quad q=\\left\\lfloor \\frac{n}{d+1}\\right\\rfloor \\ge "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.06551","kind":"arxiv","version":1},"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/2607.06551/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":"2607.06551","created_at":"2026-07-08T01:19:29.675716+00:00"},{"alias_kind":"arxiv_version","alias_value":"2607.06551v1","created_at":"2026-07-08T01:19:29.675716+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.06551","created_at":"2026-07-08T01:19:29.675716+00:00"},{"alias_kind":"pith_short_12","alias_value":"VHPUZX446DDA","created_at":"2026-07-08T01:19:29.675716+00:00"},{"alias_kind":"pith_short_16","alias_value":"VHPUZX446DDAKLUD","created_at":"2026-07-08T01:19:29.675716+00:00"},{"alias_kind":"pith_short_8","alias_value":"VHPUZX44","created_at":"2026-07-08T01:19:29.675716+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/VHPUZX446DDAKLUDTEPM7EEIUH","json":"https://pith.science/pith/VHPUZX446DDAKLUDTEPM7EEIUH.json","graph_json":"https://pith.science/api/pith-number/VHPUZX446DDAKLUDTEPM7EEIUH/graph.json","events_json":"https://pith.science/api/pith-number/VHPUZX446DDAKLUDTEPM7EEIUH/events.json","paper":"https://pith.science/paper/VHPUZX44"},"agent_actions":{"view_html":"https://pith.science/pith/VHPUZX446DDAKLUDTEPM7EEIUH","download_json":"https://pith.science/pith/VHPUZX446DDAKLUDTEPM7EEIUH.json","view_paper":"https://pith.science/paper/VHPUZX44","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2607.06551&json=true","fetch_graph":"https://pith.science/api/pith-number/VHPUZX446DDAKLUDTEPM7EEIUH/graph.json","fetch_events":"https://pith.science/api/pith-number/VHPUZX446DDAKLUDTEPM7EEIUH/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/VHPUZX446DDAKLUDTEPM7EEIUH/action/timestamp_anchor","attest_storage":"https://pith.science/pith/VHPUZX446DDAKLUDTEPM7EEIUH/action/storage_attestation","attest_author":"https://pith.science/pith/VHPUZX446DDAKLUDTEPM7EEIUH/action/author_attestation","sign_citation":"https://pith.science/pith/VHPUZX446DDAKLUDTEPM7EEIUH/action/citation_signature","submit_replication":"https://pith.science/pith/VHPUZX446DDAKLUDTEPM7EEIUH/action/replication_record"}},"created_at":"2026-07-08T01:19:29.675716+00:00","updated_at":"2026-07-08T01:19:29.675716+00:00"}