{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:CA3HIRY4754LYGGCFLPWCI5DEP","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":"4e17375f529a698e5671e97d00430b9fc933aadd30a48b9f50182b9332d2d802","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-12-27T11:22:56Z","title_canon_sha256":"9b18f289e6fb79d0a85432e8985c6e94e62a50b307f181e0c57c65652b6563b9"},"schema_version":"1.0","source":{"id":"1212.6348","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1212.6348","created_at":"2026-05-18T01:11:58Z"},{"alias_kind":"arxiv_version","alias_value":"1212.6348v2","created_at":"2026-05-18T01:11:58Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1212.6348","created_at":"2026-05-18T01:11:58Z"},{"alias_kind":"pith_short_12","alias_value":"CA3HIRY4754L","created_at":"2026-05-18T12:27:01Z"},{"alias_kind":"pith_short_16","alias_value":"CA3HIRY4754LYGGC","created_at":"2026-05-18T12:27:01Z"},{"alias_kind":"pith_short_8","alias_value":"CA3HIRY4","created_at":"2026-05-18T12:27:01Z"}],"graph_snapshots":[{"event_id":"sha256:b68689681b400df4afe1aac200ad749301af4807d884f3782d37e03971757443","target":"graph","created_at":"2026-05-18T01:11:58Z","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":"Let $G$ be an edge-colored graph. The color degree of a vertex $v$ of $G$, is defined as the number of colors of the edges incident to $v$. The color number of $G$ is defined as the number of colors of the edges in $G$. A rainbow triangle is one in which every pair of edges have distinct colors. In this paper we give some sufficient conditions for the existence of rainbow triangles in edge-colored graphs in terms of color degree, color number and edge number. As a corollary, a conjecture proposed by Li and Wang (Color degree and heterochromatic cycles in edge-colored graphs, European J. Combin","authors_text":"Binlong Li, Bo Ning, Chuandong Xu, Shenggui Zhang","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-12-27T11:22:56Z","title":"Rainbow triangles in edge-colored graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1212.6348","kind":"arxiv","version":2},"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":"sha256:74427474fbea99bd004ae0470df3959b5226a0548e734619d15d7b1b34e6098d","target":"record","created_at":"2026-05-18T01:11:58Z","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":"4e17375f529a698e5671e97d00430b9fc933aadd30a48b9f50182b9332d2d802","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-12-27T11:22:56Z","title_canon_sha256":"9b18f289e6fb79d0a85432e8985c6e94e62a50b307f181e0c57c65652b6563b9"},"schema_version":"1.0","source":{"id":"1212.6348","kind":"arxiv","version":2}},"canonical_sha256":"103674471cff78bc18c22adf6123a323dc86000111c6b7e03749b0797c2c569d","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"103674471cff78bc18c22adf6123a323dc86000111c6b7e03749b0797c2c569d","first_computed_at":"2026-05-18T01:11:58.016383Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:11:58.016383Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"me+Vsg0AlFHLCZi51/T/Tp0TUFgHTpdrgPzFlctvRwk4fyhypI8RkFZVdfDTdv7BmD8ZNEb879beW/zrZxZEBg==","signature_status":"signed_v1","signed_at":"2026-05-18T01:11:58.016716Z","signed_message":"canonical_sha256_bytes"},"source_id":"1212.6348","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:74427474fbea99bd004ae0470df3959b5226a0548e734619d15d7b1b34e6098d","sha256:b68689681b400df4afe1aac200ad749301af4807d884f3782d37e03971757443"],"state_sha256":"f26c68a9df1e0530c55ed156010d0e25ed124997a957bff98fd3f0f667efe9a0"}