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In 1963, Dirac and Erd\\H{o}s proved that $G$ contains $k$ (vertex-)disjoint cycles whenever $|H_{k}(G)| - |L_{k}(G)| \\ge k^{2} + 2k - 4$. The main result of this paper is that for $k \\ge 2$, every graph $G$ with $|V(G)| \\ge 3k$ containing at most $t$ disjoint triangles and with $|H_{k}(G)| - |L_{k}(G)| \\ge 2k + t$ contains $k$ disjoint cycles. This yields that if $k \\ge 2$ and $|H_{k}(G)| - |L_{k}(G)| \\ge 3k$, then $G$ con"},"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":"1602.02461","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-02-08T04:06:45Z","cross_cats_sorted":[],"title_canon_sha256":"3087f9b43b602d9fdbbebeb35098c5718c71e7cdddeb585997eef69ffe54e2de","abstract_canon_sha256":"2bd8b56cc8056247e9982be4fe38223a6ee4d4fd5dd5b03e71b4aa8b4f8a2874"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:21:09.865033Z","signature_b64":"UWTnJumFCmyTJVfKw4mep9XQhdOHsmDHPXmgu+of6PUE6tpMJI33gjxhsw7+Nnr3BNRa/mnMXLc41qWsqCKfBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"aad6736093b341a197ae917180ba62f5f0697f4935d10c8775e809868882bc41","last_reissued_at":"2026-05-18T01:21:09.864333Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:21:09.864333Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Strengthening theorems of Dirac and Erd\\H{o}s on disjoint cycles","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Alexandr V. Kostochka, Andrew McConvey, Henry A. Kierstead","submitted_at":"2016-02-08T04:06:45Z","abstract_excerpt":"Let $k \\ge 3$ be an integer, $H_{k}(G)$ be the set of vertices of degree at least $2k$ in a graph $G$, and $L_{k}(G)$ be the set of vertices of degree at most $2k-2$ in $G$. In 1963, Dirac and Erd\\H{o}s proved that $G$ contains $k$ (vertex-)disjoint cycles whenever $|H_{k}(G)| - |L_{k}(G)| \\ge k^{2} + 2k - 4$. The main result of this paper is that for $k \\ge 2$, every graph $G$ with $|V(G)| \\ge 3k$ containing at most $t$ disjoint triangles and with $|H_{k}(G)| - |L_{k}(G)| \\ge 2k + t$ contains $k$ disjoint cycles. This yields that if $k \\ge 2$ and $|H_{k}(G)| - |L_{k}(G)| \\ge 3k$, then $G$ con"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1602.02461","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":""},"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":"1602.02461","created_at":"2026-05-18T01:21:09.864447+00:00"},{"alias_kind":"arxiv_version","alias_value":"1602.02461v1","created_at":"2026-05-18T01:21:09.864447+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1602.02461","created_at":"2026-05-18T01:21:09.864447+00:00"},{"alias_kind":"pith_short_12","alias_value":"VLLHGYETWNA2","created_at":"2026-05-18T12:30:48.956258+00:00"},{"alias_kind":"pith_short_16","alias_value":"VLLHGYETWNA2DF5O","created_at":"2026-05-18T12:30:48.956258+00:00"},{"alias_kind":"pith_short_8","alias_value":"VLLHGYET","created_at":"2026-05-18T12:30:48.956258+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/VLLHGYETWNA2DF5OSFYYBOTC6X","json":"https://pith.science/pith/VLLHGYETWNA2DF5OSFYYBOTC6X.json","graph_json":"https://pith.science/api/pith-number/VLLHGYETWNA2DF5OSFYYBOTC6X/graph.json","events_json":"https://pith.science/api/pith-number/VLLHGYETWNA2DF5OSFYYBOTC6X/events.json","paper":"https://pith.science/paper/VLLHGYET"},"agent_actions":{"view_html":"https://pith.science/pith/VLLHGYETWNA2DF5OSFYYBOTC6X","download_json":"https://pith.science/pith/VLLHGYETWNA2DF5OSFYYBOTC6X.json","view_paper":"https://pith.science/paper/VLLHGYET","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1602.02461&json=true","fetch_graph":"https://pith.science/api/pith-number/VLLHGYETWNA2DF5OSFYYBOTC6X/graph.json","fetch_events":"https://pith.science/api/pith-number/VLLHGYETWNA2DF5OSFYYBOTC6X/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/VLLHGYETWNA2DF5OSFYYBOTC6X/action/timestamp_anchor","attest_storage":"https://pith.science/pith/VLLHGYETWNA2DF5OSFYYBOTC6X/action/storage_attestation","attest_author":"https://pith.science/pith/VLLHGYETWNA2DF5OSFYYBOTC6X/action/author_attestation","sign_citation":"https://pith.science/pith/VLLHGYETWNA2DF5OSFYYBOTC6X/action/citation_signature","submit_replication":"https://pith.science/pith/VLLHGYETWNA2DF5OSFYYBOTC6X/action/replication_record"}},"created_at":"2026-05-18T01:21:09.864447+00:00","updated_at":"2026-05-18T01:21:09.864447+00:00"}