{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:H46QIGD6L6OIOX6LW2PKRTPB2B","short_pith_number":"pith:H46QIGD6","canonical_record":{"source":{"id":"1207.3578","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-07-16T05:17:04Z","cross_cats_sorted":[],"title_canon_sha256":"6451986e7c05122f9a3d081c8a9239ba4d940a69f52b4307ac3060ffe81e78af","abstract_canon_sha256":"f5bd21dd7c8cf397fb530a654fadcc98870ca479e53c5caa43f79587afec4612"},"schema_version":"1.0"},"canonical_sha256":"3f3d04187e5f9c875fcbb69ea8cde1d0564082caa96dd4926de8f16dd38436e8","source":{"kind":"arxiv","id":"1207.3578","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1207.3578","created_at":"2026-05-18T03:51:00Z"},{"alias_kind":"arxiv_version","alias_value":"1207.3578v1","created_at":"2026-05-18T03:51:00Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1207.3578","created_at":"2026-05-18T03:51:00Z"},{"alias_kind":"pith_short_12","alias_value":"H46QIGD6L6OI","created_at":"2026-05-18T12:27:06Z"},{"alias_kind":"pith_short_16","alias_value":"H46QIGD6L6OIOX6L","created_at":"2026-05-18T12:27:06Z"},{"alias_kind":"pith_short_8","alias_value":"H46QIGD6","created_at":"2026-05-18T12:27:06Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:H46QIGD6L6OIOX6LW2PKRTPB2B","target":"record","payload":{"canonical_record":{"source":{"id":"1207.3578","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-07-16T05:17:04Z","cross_cats_sorted":[],"title_canon_sha256":"6451986e7c05122f9a3d081c8a9239ba4d940a69f52b4307ac3060ffe81e78af","abstract_canon_sha256":"f5bd21dd7c8cf397fb530a654fadcc98870ca479e53c5caa43f79587afec4612"},"schema_version":"1.0"},"canonical_sha256":"3f3d04187e5f9c875fcbb69ea8cde1d0564082caa96dd4926de8f16dd38436e8","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:51:00.715193Z","signature_b64":"OYYK7S6WtTpnSFlTBi7697w1f5EvnVUFWKF/EMCGWAVNghSqEold3bCnephBa9FQKfAzaLc5kuYQQNbo+Np+AQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3f3d04187e5f9c875fcbb69ea8cde1d0564082caa96dd4926de8f16dd38436e8","last_reissued_at":"2026-05-18T03:51:00.714469Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:51:00.714469Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1207.3578","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:51:00Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"EK9D6QcYcUTBPJ7/VghaydskKbFVHAM9nHv4gaw6LbZubYMcOdM+tv/lwxOcTt7i6S6NgZ+PiMUByy6BAOz8DA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-24T20:07:06.379892Z"},"content_sha256":"7385983da3cc16eec6b4ef9c3f01869e32377fb669bb568e07669ea8d57c8593","schema_version":"1.0","event_id":"sha256:7385983da3cc16eec6b4ef9c3f01869e32377fb669bb568e07669ea8d57c8593"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:H46QIGD6L6OIOX6LW2PKRTPB2B","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Equitable chromatic threshold of complete multipartite graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Wei Wang, Zhidan Yan","submitted_at":"2012-07-16T05:17:04Z","abstract_excerpt":"A proper vertex coloring of a graph is equitable if the sizes of color classes differ by at most one. The equitable chromatic number of a graph $G$, denoted by $\\chi_=(G)$, is the minimum $k$ such that $G$ is equitably $k$-colorable. The equitable chromatic threshold of a graph $G$, denoted by $\\chi_=^*(G)$, is the minimum $t$ such that $G$ is equitably $k$-colorable for $k\\ge t$. We develop a formula and a linear-time algorithm which compute the equitable chromatic threshold of an arbitrary complete multipartite graph."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1207.3578","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:51:00Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"4ydStL1S3vn8HQHbzRljh/iq5RazUdP9ZDyvsutb+K1WDnqdgY+EDg0fmqtY6DQM8jlI/igMh5ooJdzwIvC6AQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-24T20:07:06.380227Z"},"content_sha256":"a54fac6042cfb1a8adeebea004e0fdf0b9575d8c3a34342d0d2b25c8bc938d71","schema_version":"1.0","event_id":"sha256:a54fac6042cfb1a8adeebea004e0fdf0b9575d8c3a34342d0d2b25c8bc938d71"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/H46QIGD6L6OIOX6LW2PKRTPB2B/bundle.json","state_url":"https://pith.science/pith/H46QIGD6L6OIOX6LW2PKRTPB2B/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/H46QIGD6L6OIOX6LW2PKRTPB2B/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-24T20:07:06Z","links":{"resolver":"https://pith.science/pith/H46QIGD6L6OIOX6LW2PKRTPB2B","bundle":"https://pith.science/pith/H46QIGD6L6OIOX6LW2PKRTPB2B/bundle.json","state":"https://pith.science/pith/H46QIGD6L6OIOX6LW2PKRTPB2B/state.json","well_known_bundle":"https://pith.science/.well-known/pith/H46QIGD6L6OIOX6LW2PKRTPB2B/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:H46QIGD6L6OIOX6LW2PKRTPB2B","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":"f5bd21dd7c8cf397fb530a654fadcc98870ca479e53c5caa43f79587afec4612","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-07-16T05:17:04Z","title_canon_sha256":"6451986e7c05122f9a3d081c8a9239ba4d940a69f52b4307ac3060ffe81e78af"},"schema_version":"1.0","source":{"id":"1207.3578","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1207.3578","created_at":"2026-05-18T03:51:00Z"},{"alias_kind":"arxiv_version","alias_value":"1207.3578v1","created_at":"2026-05-18T03:51:00Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1207.3578","created_at":"2026-05-18T03:51:00Z"},{"alias_kind":"pith_short_12","alias_value":"H46QIGD6L6OI","created_at":"2026-05-18T12:27:06Z"},{"alias_kind":"pith_short_16","alias_value":"H46QIGD6L6OIOX6L","created_at":"2026-05-18T12:27:06Z"},{"alias_kind":"pith_short_8","alias_value":"H46QIGD6","created_at":"2026-05-18T12:27:06Z"}],"graph_snapshots":[{"event_id":"sha256:a54fac6042cfb1a8adeebea004e0fdf0b9575d8c3a34342d0d2b25c8bc938d71","target":"graph","created_at":"2026-05-18T03:51:00Z","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 proper vertex coloring of a graph is equitable if the sizes of color classes differ by at most one. The equitable chromatic number of a graph $G$, denoted by $\\chi_=(G)$, is the minimum $k$ such that $G$ is equitably $k$-colorable. The equitable chromatic threshold of a graph $G$, denoted by $\\chi_=^*(G)$, is the minimum $t$ such that $G$ is equitably $k$-colorable for $k\\ge t$. We develop a formula and a linear-time algorithm which compute the equitable chromatic threshold of an arbitrary complete multipartite graph.","authors_text":"Wei Wang, Zhidan Yan","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-07-16T05:17:04Z","title":"Equitable chromatic threshold of complete multipartite graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1207.3578","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":"sha256:7385983da3cc16eec6b4ef9c3f01869e32377fb669bb568e07669ea8d57c8593","target":"record","created_at":"2026-05-18T03:51:00Z","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":"f5bd21dd7c8cf397fb530a654fadcc98870ca479e53c5caa43f79587afec4612","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-07-16T05:17:04Z","title_canon_sha256":"6451986e7c05122f9a3d081c8a9239ba4d940a69f52b4307ac3060ffe81e78af"},"schema_version":"1.0","source":{"id":"1207.3578","kind":"arxiv","version":1}},"canonical_sha256":"3f3d04187e5f9c875fcbb69ea8cde1d0564082caa96dd4926de8f16dd38436e8","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"3f3d04187e5f9c875fcbb69ea8cde1d0564082caa96dd4926de8f16dd38436e8","first_computed_at":"2026-05-18T03:51:00.714469Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:51:00.714469Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"OYYK7S6WtTpnSFlTBi7697w1f5EvnVUFWKF/EMCGWAVNghSqEold3bCnephBa9FQKfAzaLc5kuYQQNbo+Np+AQ==","signature_status":"signed_v1","signed_at":"2026-05-18T03:51:00.715193Z","signed_message":"canonical_sha256_bytes"},"source_id":"1207.3578","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:7385983da3cc16eec6b4ef9c3f01869e32377fb669bb568e07669ea8d57c8593","sha256:a54fac6042cfb1a8adeebea004e0fdf0b9575d8c3a34342d0d2b25c8bc938d71"],"state_sha256":"a7084d934d93f7bbfa4c06851632b4ab49f0ac05508a58c4272f1d1963ed2768"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"u43Hr7KSgK1JgeyOuHSZbw3hVS644DzU+xtqmIsw1mOjUoXKB+LIcg8uYSuhFrQY20rptVGlsxn9D1bHSOn/Aw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-24T20:07:06.382041Z","bundle_sha256":"5bb519bacd43569a8876a5024d66d8042545353b8696588e7b62769dc1953ee3"}}