{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2024:HGKDH622MM3KOZF5UCEDOFIC2C","short_pith_number":"pith:HGKDH622","schema_version":"1.0","canonical_sha256":"399433fb5a6336a764bda088371502d089852a197a0315951ae1624e6d63722d","source":{"kind":"arxiv","id":"2408.05547","version":1},"attestation_state":"computed","paper":{"title":"Triangle-free Graphs with Large Minimum Common Degree","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Fan Zhao, Jian Wang, Weihua Yang","submitted_at":"2024-08-10T13:12:27Z","abstract_excerpt":"Let $G$ be a graph. For $x\\in V(G)$, let $N(x)=\\{y\\in V(G)\\colon xy\\in E(G)\\}$. The minimum common degree of $G$, denoted by $\\delta_{2}(G)$, is defined as the minimum of $|N(x)\\cap N(y)|$ over all non-edges $xy$ of $G$. In 1982, H\\\"{a}ggkvist showed that every triangle-free graph with minimum degree greater than $\\lfloor\\frac{3n}{8}\\rfloor$ is homomorphic to a cycle of length 5. In this paper, we prove that every triangle-free graph with minimum common degree greater than $\\lfloor\\frac{n}{8}\\rfloor$ is homomorphic to a cycle of length 5, which implies H\\\"{a}ggkvist's result. The balanced blow"},"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":"2408.05547","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2024-08-10T13:12:27Z","cross_cats_sorted":[],"title_canon_sha256":"9d7bd3b85d93e79cfd3f76c010b427894d65b05c27350263fc34b2bc36fd3a77","abstract_canon_sha256":"37b2b98f2eee25f899ff7e7fae248dd9d55a841fc5b0146d931ec8096d5f4ee6"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T08:54:21.670715Z","signature_b64":"tgS4O/NK9CXUpfZciDkl/WX2honAsGlmMXvpQ9sG9opFSg91r3SlgSZTlje5egmeLy+QL3dHzGnVIVkK6FV/Aw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"399433fb5a6336a764bda088371502d089852a197a0315951ae1624e6d63722d","last_reissued_at":"2026-07-05T08:54:21.670274Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T08:54:21.670274Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Triangle-free Graphs with Large Minimum Common Degree","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Fan Zhao, Jian Wang, Weihua Yang","submitted_at":"2024-08-10T13:12:27Z","abstract_excerpt":"Let $G$ be a graph. For $x\\in V(G)$, let $N(x)=\\{y\\in V(G)\\colon xy\\in E(G)\\}$. The minimum common degree of $G$, denoted by $\\delta_{2}(G)$, is defined as the minimum of $|N(x)\\cap N(y)|$ over all non-edges $xy$ of $G$. In 1982, H\\\"{a}ggkvist showed that every triangle-free graph with minimum degree greater than $\\lfloor\\frac{3n}{8}\\rfloor$ is homomorphic to a cycle of length 5. In this paper, we prove that every triangle-free graph with minimum common degree greater than $\\lfloor\\frac{n}{8}\\rfloor$ is homomorphic to a cycle of length 5, which implies H\\\"{a}ggkvist's result. The balanced blow"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2408.05547","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/2408.05547/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":"2408.05547","created_at":"2026-07-05T08:54:21.670362+00:00"},{"alias_kind":"arxiv_version","alias_value":"2408.05547v1","created_at":"2026-07-05T08:54:21.670362+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2408.05547","created_at":"2026-07-05T08:54:21.670362+00:00"},{"alias_kind":"pith_short_12","alias_value":"HGKDH622MM3K","created_at":"2026-07-05T08:54:21.670362+00:00"},{"alias_kind":"pith_short_16","alias_value":"HGKDH622MM3KOZF5","created_at":"2026-07-05T08:54:21.670362+00:00"},{"alias_kind":"pith_short_8","alias_value":"HGKDH622","created_at":"2026-07-05T08:54:21.670362+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/HGKDH622MM3KOZF5UCEDOFIC2C","json":"https://pith.science/pith/HGKDH622MM3KOZF5UCEDOFIC2C.json","graph_json":"https://pith.science/api/pith-number/HGKDH622MM3KOZF5UCEDOFIC2C/graph.json","events_json":"https://pith.science/api/pith-number/HGKDH622MM3KOZF5UCEDOFIC2C/events.json","paper":"https://pith.science/paper/HGKDH622"},"agent_actions":{"view_html":"https://pith.science/pith/HGKDH622MM3KOZF5UCEDOFIC2C","download_json":"https://pith.science/pith/HGKDH622MM3KOZF5UCEDOFIC2C.json","view_paper":"https://pith.science/paper/HGKDH622","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2408.05547&json=true","fetch_graph":"https://pith.science/api/pith-number/HGKDH622MM3KOZF5UCEDOFIC2C/graph.json","fetch_events":"https://pith.science/api/pith-number/HGKDH622MM3KOZF5UCEDOFIC2C/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/HGKDH622MM3KOZF5UCEDOFIC2C/action/timestamp_anchor","attest_storage":"https://pith.science/pith/HGKDH622MM3KOZF5UCEDOFIC2C/action/storage_attestation","attest_author":"https://pith.science/pith/HGKDH622MM3KOZF5UCEDOFIC2C/action/author_attestation","sign_citation":"https://pith.science/pith/HGKDH622MM3KOZF5UCEDOFIC2C/action/citation_signature","submit_replication":"https://pith.science/pith/HGKDH622MM3KOZF5UCEDOFIC2C/action/replication_record"}},"created_at":"2026-07-05T08:54:21.670362+00:00","updated_at":"2026-07-05T08:54:21.670362+00:00"}