{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:RKR6GLIWLBOZ2DEJJEKAG74FEO","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":"36bdfac7198dd4bb5ca4067599ff271de4d4d18e6d298fc5fb8f66e91c53199a","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-09-18T13:30:17Z","title_canon_sha256":"76bb039b1dba33a23fe12e08a125d70180777549100ec5de25569466ac97b03a"},"schema_version":"1.0","source":{"id":"1509.05621","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1509.05621","created_at":"2026-05-18T01:32:42Z"},{"alias_kind":"arxiv_version","alias_value":"1509.05621v1","created_at":"2026-05-18T01:32:42Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1509.05621","created_at":"2026-05-18T01:32:42Z"},{"alias_kind":"pith_short_12","alias_value":"RKR6GLIWLBOZ","created_at":"2026-05-18T12:29:39Z"},{"alias_kind":"pith_short_16","alias_value":"RKR6GLIWLBOZ2DEJ","created_at":"2026-05-18T12:29:39Z"},{"alias_kind":"pith_short_8","alias_value":"RKR6GLIW","created_at":"2026-05-18T12:29:39Z"}],"graph_snapshots":[{"event_id":"sha256:5727c597b51ebf078b5951d9ee8c7cec184c358af4e52291ae366331690f24cb","target":"graph","created_at":"2026-05-18T01:32:42Z","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":"A colored graph is a complete graph in which a color has been assigned to each edge, and a colorful cycle is a cycle in which each edge has a different color. We first show that a colored graph lacks colorful cycles iff it is Gallai, i.e., lacks colorful triangles. We then show that, under the operation $m\\circ n\\equiv m+n-2$, the omitted lengths of colorful cycles in a colored graph form a monoid isomorphic to a submonoid of the natural numbers which contains all integers past some point. We prove that several but not all such monoids are realized.\n  We then characterize exact Gallai graphs, ","authors_text":"Ale\\v{s} Pultr, Petr Vojt\\v{e}chovsk\\'y, Richard N. Ball","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-09-18T13:30:17Z","title":"Colored graphs without colorful cycles"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1509.05621","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":"sha256:9d58e0d81de0bc2eb803bd0f9f033bde125ea16ea4eb2844b21230bf468801b2","target":"record","created_at":"2026-05-18T01:32:42Z","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":"36bdfac7198dd4bb5ca4067599ff271de4d4d18e6d298fc5fb8f66e91c53199a","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-09-18T13:30:17Z","title_canon_sha256":"76bb039b1dba33a23fe12e08a125d70180777549100ec5de25569466ac97b03a"},"schema_version":"1.0","source":{"id":"1509.05621","kind":"arxiv","version":1}},"canonical_sha256":"8aa3e32d16585d9d0c894914037f8523bff0a4cf28cec5e21019d77680a08a5b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"8aa3e32d16585d9d0c894914037f8523bff0a4cf28cec5e21019d77680a08a5b","first_computed_at":"2026-05-18T01:32:42.739576Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:32:42.739576Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"XTtHEJZw9AYaVX9QsNR3bg7ApdLtauKQaX3qqWNt//uNcdfaT0wm6QtNQX9hEEyPQeBTqO4UK3eLe/qFcl29Dw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:32:42.740179Z","signed_message":"canonical_sha256_bytes"},"source_id":"1509.05621","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:9d58e0d81de0bc2eb803bd0f9f033bde125ea16ea4eb2844b21230bf468801b2","sha256:5727c597b51ebf078b5951d9ee8c7cec184c358af4e52291ae366331690f24cb"],"state_sha256":"e24984ce00ab9557e5c07484fb08a9a7dad9828909a453c03c746befb40057d3"}