{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:YWDNCCKGCGA2X6ZTLZSDXD6LL5","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":"60dc5e81d83c366dac00527bb28b7c58d3f9ce5b8289488f401b911b464ea55a","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2012-10-04T08:46:25Z","title_canon_sha256":"ef22ed4b558fc2afc12f466616129d3c460444840fa1f504c52c7447cf19c662"},"schema_version":"1.0","source":{"id":"1210.1336","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1210.1336","created_at":"2026-05-18T03:44:03Z"},{"alias_kind":"arxiv_version","alias_value":"1210.1336v1","created_at":"2026-05-18T03:44:03Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1210.1336","created_at":"2026-05-18T03:44:03Z"},{"alias_kind":"pith_short_12","alias_value":"YWDNCCKGCGA2","created_at":"2026-05-18T12:27:30Z"},{"alias_kind":"pith_short_16","alias_value":"YWDNCCKGCGA2X6ZT","created_at":"2026-05-18T12:27:30Z"},{"alias_kind":"pith_short_8","alias_value":"YWDNCCKG","created_at":"2026-05-18T12:27:30Z"}],"graph_snapshots":[{"event_id":"sha256:c4b885a4b8f37db49e4bdfe123e77c277c368735ec127f50dafedcfef8326491","target":"graph","created_at":"2026-05-18T03:44:03Z","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":"In this note, we give some necessary conditions for an $r$-partite graph such that the edge ring of the graph is Cohen-Macaulay. It is proved that if $G$ is an $r$-partite Cohen-Macaulay graph which is covered by some disjoint cliques of size $r$, then the clique cover is unique.","authors_text":"Asghar Madadi, Rashid Zaare-Nahandi","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2012-10-04T08:46:25Z","title":"Cohen-Macaulay $r$-partite graphs with minimal clique cover"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1210.1336","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":"sha256:68cce131c36272aff20227754364fd2666f02566f9111f57cc1c237f219f8da4","target":"record","created_at":"2026-05-18T03:44:03Z","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":"60dc5e81d83c366dac00527bb28b7c58d3f9ce5b8289488f401b911b464ea55a","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2012-10-04T08:46:25Z","title_canon_sha256":"ef22ed4b558fc2afc12f466616129d3c460444840fa1f504c52c7447cf19c662"},"schema_version":"1.0","source":{"id":"1210.1336","kind":"arxiv","version":1}},"canonical_sha256":"c586d109461181abfb335e643b8fcb5f7c1daeb7aee97b650a5750e4ea4fd12d","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"c586d109461181abfb335e643b8fcb5f7c1daeb7aee97b650a5750e4ea4fd12d","first_computed_at":"2026-05-18T03:44:03.878674Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:44:03.878674Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"SWH6kFTIDPxdnfZrHmCP6qTR2cPFtlVAv+GrUdCTwrEQu/+j9hZYCGueEoCbkHjShL64eE2b66NN/0lJF7OuCQ==","signature_status":"signed_v1","signed_at":"2026-05-18T03:44:03.879378Z","signed_message":"canonical_sha256_bytes"},"source_id":"1210.1336","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:68cce131c36272aff20227754364fd2666f02566f9111f57cc1c237f219f8da4","sha256:c4b885a4b8f37db49e4bdfe123e77c277c368735ec127f50dafedcfef8326491"],"state_sha256":"ec7368b98ff37012c7d905d51ec43e38eb37850dfca466c75ec2c60b9633489c"}