{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2024:HGKDH622MM3KOZF5UCEDOFIC2C","short_pith_number":"pith:HGKDH622","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"},"canonical_sha256":"399433fb5a6336a764bda088371502d089852a197a0315951ae1624e6d63722d","source":{"kind":"arxiv","id":"2408.05547","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2408.05547","created_at":"2026-07-05T08:54:21Z"},{"alias_kind":"arxiv_version","alias_value":"2408.05547v1","created_at":"2026-07-05T08:54:21Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2408.05547","created_at":"2026-07-05T08:54:21Z"},{"alias_kind":"pith_short_12","alias_value":"HGKDH622MM3K","created_at":"2026-07-05T08:54:21Z"},{"alias_kind":"pith_short_16","alias_value":"HGKDH622MM3KOZF5","created_at":"2026-07-05T08:54:21Z"},{"alias_kind":"pith_short_8","alias_value":"HGKDH622","created_at":"2026-07-05T08:54:21Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2024:HGKDH622MM3KOZF5UCEDOFIC2C","target":"record","payload":{"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"},"canonical_sha256":"399433fb5a6336a764bda088371502d089852a197a0315951ae1624e6d63722d","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"},"source_kind":"arxiv","source_id":"2408.05547","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T08:54:21Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"g/K2WLnhIhCSoT/O6Aqycl0BXpSt39fQdQCRog4fx+kfYu2qCpBjnglGK6wr5oJqhioNeLf/ng/UeMRjWrOaDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-05T15:15:28.099700Z"},"content_sha256":"a090f9e34ed4bb196f91967d3830f02aed5c495af4247e49687f8c168313d9e0","schema_version":"1.0","event_id":"sha256:a090f9e34ed4bb196f91967d3830f02aed5c495af4247e49687f8c168313d9e0"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2024:HGKDH622MM3KOZF5UCEDOFIC2C","target":"graph","payload":{"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T08:54:21Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"c7S9mTJ5iGdojxQvE7McODAJp5mKFP4xQntkDkgW2G189Rz/09EPWvXAZIL7V9q6MubvCpJKweupGG/J8U29DA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-05T15:15:28.100579Z"},"content_sha256":"5a4b5f1ca2e05ac0bc04dac2318651e09bdb176a0783de09d1b20deb62881f87","schema_version":"1.0","event_id":"sha256:5a4b5f1ca2e05ac0bc04dac2318651e09bdb176a0783de09d1b20deb62881f87"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/HGKDH622MM3KOZF5UCEDOFIC2C/bundle.json","state_url":"https://pith.science/pith/HGKDH622MM3KOZF5UCEDOFIC2C/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/HGKDH622MM3KOZF5UCEDOFIC2C/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-08-05T15:15:28Z","links":{"resolver":"https://pith.science/pith/HGKDH622MM3KOZF5UCEDOFIC2C","bundle":"https://pith.science/pith/HGKDH622MM3KOZF5UCEDOFIC2C/bundle.json","state":"https://pith.science/pith/HGKDH622MM3KOZF5UCEDOFIC2C/state.json","well_known_bundle":"https://pith.science/.well-known/pith/HGKDH622MM3KOZF5UCEDOFIC2C/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2024:HGKDH622MM3KOZF5UCEDOFIC2C","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":"37b2b98f2eee25f899ff7e7fae248dd9d55a841fc5b0146d931ec8096d5f4ee6","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2024-08-10T13:12:27Z","title_canon_sha256":"9d7bd3b85d93e79cfd3f76c010b427894d65b05c27350263fc34b2bc36fd3a77"},"schema_version":"1.0","source":{"id":"2408.05547","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2408.05547","created_at":"2026-07-05T08:54:21Z"},{"alias_kind":"arxiv_version","alias_value":"2408.05547v1","created_at":"2026-07-05T08:54:21Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2408.05547","created_at":"2026-07-05T08:54:21Z"},{"alias_kind":"pith_short_12","alias_value":"HGKDH622MM3K","created_at":"2026-07-05T08:54:21Z"},{"alias_kind":"pith_short_16","alias_value":"HGKDH622MM3KOZF5","created_at":"2026-07-05T08:54:21Z"},{"alias_kind":"pith_short_8","alias_value":"HGKDH622","created_at":"2026-07-05T08:54:21Z"}],"graph_snapshots":[{"event_id":"sha256:5a4b5f1ca2e05ac0bc04dac2318651e09bdb176a0783de09d1b20deb62881f87","target":"graph","created_at":"2026-07-05T08:54:21Z","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/2408.05547/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"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","authors_text":"Fan Zhao, Jian Wang, Weihua Yang","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2024-08-10T13:12:27Z","title":"Triangle-free Graphs with Large Minimum Common Degree"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2408.05547","kind":"arxiv","version":1},"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:a090f9e34ed4bb196f91967d3830f02aed5c495af4247e49687f8c168313d9e0","target":"record","created_at":"2026-07-05T08:54:21Z","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":"37b2b98f2eee25f899ff7e7fae248dd9d55a841fc5b0146d931ec8096d5f4ee6","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2024-08-10T13:12:27Z","title_canon_sha256":"9d7bd3b85d93e79cfd3f76c010b427894d65b05c27350263fc34b2bc36fd3a77"},"schema_version":"1.0","source":{"id":"2408.05547","kind":"arxiv","version":1}},"canonical_sha256":"399433fb5a6336a764bda088371502d089852a197a0315951ae1624e6d63722d","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"399433fb5a6336a764bda088371502d089852a197a0315951ae1624e6d63722d","first_computed_at":"2026-07-05T08:54:21.670274Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T08:54:21.670274Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"tgS4O/NK9CXUpfZciDkl/WX2honAsGlmMXvpQ9sG9opFSg91r3SlgSZTlje5egmeLy+QL3dHzGnVIVkK6FV/Aw==","signature_status":"signed_v1","signed_at":"2026-07-05T08:54:21.670715Z","signed_message":"canonical_sha256_bytes"},"source_id":"2408.05547","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:a090f9e34ed4bb196f91967d3830f02aed5c495af4247e49687f8c168313d9e0","sha256:5a4b5f1ca2e05ac0bc04dac2318651e09bdb176a0783de09d1b20deb62881f87"],"state_sha256":"037bb5163eac6c05a20b5e59f8d64351d177413134d7cb6c5a37b3acb7cdd49a"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"X9Oe08XIl9eIpiL1HCpuq4OOAg47a/YG7s1kYC7IoexNBdMNKfLmKwIzAwrnHhY2O5h8T/AOKPCH4f9jHvpwCQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-05T15:15:28.108209Z","bundle_sha256":"d7edd593c71cf7b5e522691e477bd95b7c65f29ee600ea9aafffe3c71f612148"}}