{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2015:RKR6GLIWLBOZ2DEJJEKAG74FEO","short_pith_number":"pith:RKR6GLIW","schema_version":"1.0","canonical_sha256":"8aa3e32d16585d9d0c894914037f8523bff0a4cf28cec5e21019d77680a08a5b","source":{"kind":"arxiv","id":"1509.05621","version":1},"attestation_state":"computed","paper":{"title":"Colored graphs without colorful cycles","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Ale\\v{s} Pultr, Petr Vojt\\v{e}chovsk\\'y, Richard N. Ball","submitted_at":"2015-09-18T13:30:17Z","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, "},"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":"1509.05621","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-09-18T13:30:17Z","cross_cats_sorted":[],"title_canon_sha256":"76bb039b1dba33a23fe12e08a125d70180777549100ec5de25569466ac97b03a","abstract_canon_sha256":"36bdfac7198dd4bb5ca4067599ff271de4d4d18e6d298fc5fb8f66e91c53199a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:32:42.740179Z","signature_b64":"XTtHEJZw9AYaVX9QsNR3bg7ApdLtauKQaX3qqWNt//uNcdfaT0wm6QtNQX9hEEyPQeBTqO4UK3eLe/qFcl29Dw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8aa3e32d16585d9d0c894914037f8523bff0a4cf28cec5e21019d77680a08a5b","last_reissued_at":"2026-05-18T01:32:42.739576Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:32:42.739576Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Colored graphs without colorful cycles","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Ale\\v{s} Pultr, Petr Vojt\\v{e}chovsk\\'y, Richard N. Ball","submitted_at":"2015-09-18T13:30:17Z","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, "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1509.05621","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":""},"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":"1509.05621","created_at":"2026-05-18T01:32:42.739667+00:00"},{"alias_kind":"arxiv_version","alias_value":"1509.05621v1","created_at":"2026-05-18T01:32:42.739667+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1509.05621","created_at":"2026-05-18T01:32:42.739667+00:00"},{"alias_kind":"pith_short_12","alias_value":"RKR6GLIWLBOZ","created_at":"2026-05-18T12:29:39.896362+00:00"},{"alias_kind":"pith_short_16","alias_value":"RKR6GLIWLBOZ2DEJ","created_at":"2026-05-18T12:29:39.896362+00:00"},{"alias_kind":"pith_short_8","alias_value":"RKR6GLIW","created_at":"2026-05-18T12:29:39.896362+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/RKR6GLIWLBOZ2DEJJEKAG74FEO","json":"https://pith.science/pith/RKR6GLIWLBOZ2DEJJEKAG74FEO.json","graph_json":"https://pith.science/api/pith-number/RKR6GLIWLBOZ2DEJJEKAG74FEO/graph.json","events_json":"https://pith.science/api/pith-number/RKR6GLIWLBOZ2DEJJEKAG74FEO/events.json","paper":"https://pith.science/paper/RKR6GLIW"},"agent_actions":{"view_html":"https://pith.science/pith/RKR6GLIWLBOZ2DEJJEKAG74FEO","download_json":"https://pith.science/pith/RKR6GLIWLBOZ2DEJJEKAG74FEO.json","view_paper":"https://pith.science/paper/RKR6GLIW","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1509.05621&json=true","fetch_graph":"https://pith.science/api/pith-number/RKR6GLIWLBOZ2DEJJEKAG74FEO/graph.json","fetch_events":"https://pith.science/api/pith-number/RKR6GLIWLBOZ2DEJJEKAG74FEO/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/RKR6GLIWLBOZ2DEJJEKAG74FEO/action/timestamp_anchor","attest_storage":"https://pith.science/pith/RKR6GLIWLBOZ2DEJJEKAG74FEO/action/storage_attestation","attest_author":"https://pith.science/pith/RKR6GLIWLBOZ2DEJJEKAG74FEO/action/author_attestation","sign_citation":"https://pith.science/pith/RKR6GLIWLBOZ2DEJJEKAG74FEO/action/citation_signature","submit_replication":"https://pith.science/pith/RKR6GLIWLBOZ2DEJJEKAG74FEO/action/replication_record"}},"created_at":"2026-05-18T01:32:42.739667+00:00","updated_at":"2026-05-18T01:32:42.739667+00:00"}