{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:DVE343FNGNDT2LOJVFGR7GDGDZ","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":"73c5c6e1f956cf6b5307425467eb4fae51b893958945109b37f198c747fe2883","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-04-17T08:25:36Z","title_canon_sha256":"ed5836939ae2ccf93caa425aa657ee44dbce78ca29893a6448c3e1dda36b8473"},"schema_version":"1.0","source":{"id":"1904.08126","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1904.08126","created_at":"2026-07-05T02:09:34Z"},{"alias_kind":"arxiv_version","alias_value":"1904.08126v3","created_at":"2026-07-05T02:09:34Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1904.08126","created_at":"2026-07-05T02:09:34Z"},{"alias_kind":"pith_short_12","alias_value":"DVE343FNGNDT","created_at":"2026-07-05T02:09:34Z"},{"alias_kind":"pith_short_16","alias_value":"DVE343FNGNDT2LOJ","created_at":"2026-07-05T02:09:34Z"},{"alias_kind":"pith_short_8","alias_value":"DVE343FN","created_at":"2026-07-05T02:09:34Z"}],"graph_snapshots":[{"event_id":"sha256:403b8090016f85113067b7c0778b7d8e1c5a74e998a4d6c5c42ad2ea8a5dbf1d","target":"graph","created_at":"2026-07-05T02:09:34Z","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/1904.08126/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"In this paper, we prove a tight minimum degree condition in general graphs for the existence of paths between two given endpoints, whose lengths form a long arithmetic progression with common difference one or two. This allows us to obtain a number of exact and optimal results on cycle lengths in graphs of given minimum degree, connectivity or chromatic number.\n  More precisely, we prove the following statements by a unified approach. (1) Every graph $G$ with minimum degree at least $k+1$ contains cycles of all even lengths modulo $k$; in addition, if $G$ is 2-connected and non-bipartite, then","authors_text":"Chun-Hung Liu, Jie Ma, Jun Gao, Qingyi Huo","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-04-17T08:25:36Z","title":"A unified proof of conjectures on cycle lengths in graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1904.08126","kind":"arxiv","version":3},"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":"sha256:02dcbfd7d0309ce7d4f7119b66c0e6a9aaf5f5cadca4f4340f1c756dad5395b8","target":"record","created_at":"2026-07-05T02:09:34Z","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":"73c5c6e1f956cf6b5307425467eb4fae51b893958945109b37f198c747fe2883","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-04-17T08:25:36Z","title_canon_sha256":"ed5836939ae2ccf93caa425aa657ee44dbce78ca29893a6448c3e1dda36b8473"},"schema_version":"1.0","source":{"id":"1904.08126","kind":"arxiv","version":3}},"canonical_sha256":"1d49be6cad33473d2dc9a94d1f98661e522be7ea6e9c4a6355284b3c6e1e9c22","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"1d49be6cad33473d2dc9a94d1f98661e522be7ea6e9c4a6355284b3c6e1e9c22","first_computed_at":"2026-07-05T02:09:34.471618Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T02:09:34.471618Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"padde1OC9/lE+QAuf8sJNykjcQj614fsp+EaEJ4fOv1b1MAGwj2MsRT7cbg4MsJqPIoN5JCxkAa0O0cAjTPfCw==","signature_status":"signed_v1","signed_at":"2026-07-05T02:09:34.472033Z","signed_message":"canonical_sha256_bytes"},"source_id":"1904.08126","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:02dcbfd7d0309ce7d4f7119b66c0e6a9aaf5f5cadca4f4340f1c756dad5395b8","sha256:403b8090016f85113067b7c0778b7d8e1c5a74e998a4d6c5c42ad2ea8a5dbf1d"],"state_sha256":"934048585c7c1d54d7943b8e9cf7b5371c0e8a152da1444763b622c535a300f6"}