{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:2YOEFJUZ5MFD4J22HCRLBCULNR","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":"31b2acfd12449f5d12800c9cbcae83e5a6ba3d1986d67b96b738f00a614daa2b","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-10-02T01:32:19Z","title_canon_sha256":"a9e321565c6a118c4ed4e1f1a684a237735b64b04694aa3839d3bbbd7fef2454"},"schema_version":"1.0","source":{"id":"1410.0430","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1410.0430","created_at":"2026-05-18T02:41:16Z"},{"alias_kind":"arxiv_version","alias_value":"1410.0430v1","created_at":"2026-05-18T02:41:16Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1410.0430","created_at":"2026-05-18T02:41:16Z"},{"alias_kind":"pith_short_12","alias_value":"2YOEFJUZ5MFD","created_at":"2026-05-18T12:28:11Z"},{"alias_kind":"pith_short_16","alias_value":"2YOEFJUZ5MFD4J22","created_at":"2026-05-18T12:28:11Z"},{"alias_kind":"pith_short_8","alias_value":"2YOEFJUZ","created_at":"2026-05-18T12:28:11Z"}],"graph_snapshots":[{"event_id":"sha256:c509620e16fe113d09af79f30542ef9623209a61b30497379656e01162c9569a","target":"graph","created_at":"2026-05-18T02:41:16Z","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"},"paper":{"abstract_excerpt":"It is proved that there exists an absolute constant c > 0 such that for every natural number k, every non-bipartite 2-connected graph with average degree at least ck contains k cycles with consecutive odd lengths. This implies the existence of the absolute constant d > 0 that every non-bipartite 2-connected graph with minimum degree at least dk contains cycles of all lengths modulo k, thus providing an answer (in a strong form) to a question of Thomassen. Both results are sharp up to the constant factors.","authors_text":"Jie Ma","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-10-02T01:32:19Z","title":"Cycles with consecutive odd lengths"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1410.0430","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":"sha256:243b486fcadf46672a24d96a557d5243f85969cf81b578f752ebc706ab0ff6b8","target":"record","created_at":"2026-05-18T02:41:16Z","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":"31b2acfd12449f5d12800c9cbcae83e5a6ba3d1986d67b96b738f00a614daa2b","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-10-02T01:32:19Z","title_canon_sha256":"a9e321565c6a118c4ed4e1f1a684a237735b64b04694aa3839d3bbbd7fef2454"},"schema_version":"1.0","source":{"id":"1410.0430","kind":"arxiv","version":1}},"canonical_sha256":"d61c42a699eb0a3e275a38a2b08a8b6c4bc649436ab76873646511e09d81273b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"d61c42a699eb0a3e275a38a2b08a8b6c4bc649436ab76873646511e09d81273b","first_computed_at":"2026-05-18T02:41:16.442827Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:41:16.442827Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"YHdjUbcTG+42iavu/0YD95AHsJdjskoHVZILAD3vGY28LmEVt1oSwdYBHO3dMFoFr2UutMbiSW5U1mu5Zb8DBA==","signature_status":"signed_v1","signed_at":"2026-05-18T02:41:16.443274Z","signed_message":"canonical_sha256_bytes"},"source_id":"1410.0430","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:243b486fcadf46672a24d96a557d5243f85969cf81b578f752ebc706ab0ff6b8","sha256:c509620e16fe113d09af79f30542ef9623209a61b30497379656e01162c9569a"],"state_sha256":"cec3e16ba434ab549e550b822537a7806786e17003e1844cac0c45cba9719633"}