{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:2YBSQ4HGAA27EPNIYQASEHSHIZ","short_pith_number":"pith:2YBSQ4HG","canonical_record":{"source":{"id":"1203.1967","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2012-03-09T01:12:54Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"f7c38f4d8bcea3d40a9d339cdcb4e922babb6df85499d9cb94e54246baab5c81","abstract_canon_sha256":"18eb1ef8bf3e0e66e4ff6c88bdf69dd13e7807e9dead304aa840136636ecc821"},"schema_version":"1.0"},"canonical_sha256":"d6032870e60035f23da8c401221e474654093034a0fcfe81c29566e6c11eb720","source":{"kind":"arxiv","id":"1203.1967","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1203.1967","created_at":"2026-05-18T04:00:27Z"},{"alias_kind":"arxiv_version","alias_value":"1203.1967v1","created_at":"2026-05-18T04:00:27Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1203.1967","created_at":"2026-05-18T04:00:27Z"},{"alias_kind":"pith_short_12","alias_value":"2YBSQ4HGAA27","created_at":"2026-05-18T12:26:50Z"},{"alias_kind":"pith_short_16","alias_value":"2YBSQ4HGAA27EPNI","created_at":"2026-05-18T12:26:50Z"},{"alias_kind":"pith_short_8","alias_value":"2YBSQ4HG","created_at":"2026-05-18T12:26:50Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:2YBSQ4HGAA27EPNIYQASEHSHIZ","target":"record","payload":{"canonical_record":{"source":{"id":"1203.1967","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2012-03-09T01:12:54Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"f7c38f4d8bcea3d40a9d339cdcb4e922babb6df85499d9cb94e54246baab5c81","abstract_canon_sha256":"18eb1ef8bf3e0e66e4ff6c88bdf69dd13e7807e9dead304aa840136636ecc821"},"schema_version":"1.0"},"canonical_sha256":"d6032870e60035f23da8c401221e474654093034a0fcfe81c29566e6c11eb720","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:00:27.881489Z","signature_b64":"38j9UOisIyyu01PoJU79oLNpP8blb5SGHCYpwMntXoJ0d4Ln5c2/s4fzHtojt0mlQPJTSQYcynScQ9i6J0TeBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d6032870e60035f23da8c401221e474654093034a0fcfe81c29566e6c11eb720","last_reissued_at":"2026-05-18T04:00:27.880796Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:00:27.880796Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1203.1967","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":"2026-05-18T04:00:27Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"54R/TyEs8t2TltobohwcpFhdsmP1Y2w34qv0+msAqMDr/3bNcBQSxcIMzbSMV2+67j7BJsqxJ2dxTV5Dm2koAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-06T03:01:37.852458Z"},"content_sha256":"fedf4b4edb4a63caeaa87704eb4ba25c2c550655f616bddabbf5524184b9b729","schema_version":"1.0","event_id":"sha256:fedf4b4edb4a63caeaa87704eb4ba25c2c550655f616bddabbf5524184b9b729"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:2YBSQ4HGAA27EPNIYQASEHSHIZ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Cohen--Macaulaynees for symbolic power ideals of edge ideals","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.AC","authors_text":"Giancarlo Rinaldo, Ken-ichi Yoshida, Naoki Terai","submitted_at":"2012-03-09T01:12:54Z","abstract_excerpt":"Let $S = K[x_1,..., x_n]$ be a polynomial ring over a field $K$. Let $I(G) \\subseteq S$ denote the edge ideal of a graph $G$. We show that the $\\ell$th symbolic power $I(G)^{(\\ell)}$ is a Cohen-Macaulay ideal (i.e., $S/I(G)^{(\\ell)}$ is Cohen-Macaulay) for some integer $\\ell \\ge 3$ if and only if $G$ is a disjoint union of finitely many complete graphs. When this is the case, all the symbolic powers $I(G)^{(\\ell)}$ are Cohen-Macaulay ideals. Similarly, we characterize graphs $G$ for which $S/I(G)^{(\\ell)}$ has (FLC).\n  As an application, we show that an edge ideal $I(G)$ is complete intersecti"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1203.1967","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":"2026-05-18T04:00:27Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Lc1gfz2rw5qXpFY59A2/wwChfPVVWWLx6f0Z++jwFJ//jHqOrgkzHlze8f/2ClwRVLdavlHwF/J8dJ8WWALIDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-06T03:01:37.853109Z"},"content_sha256":"89633f3ab1790280829815da90443a0b9065c28823a5fe07a830bf5554719ffa","schema_version":"1.0","event_id":"sha256:89633f3ab1790280829815da90443a0b9065c28823a5fe07a830bf5554719ffa"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/2YBSQ4HGAA27EPNIYQASEHSHIZ/bundle.json","state_url":"https://pith.science/pith/2YBSQ4HGAA27EPNIYQASEHSHIZ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/2YBSQ4HGAA27EPNIYQASEHSHIZ/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":"2026-06-06T03:01:37Z","links":{"resolver":"https://pith.science/pith/2YBSQ4HGAA27EPNIYQASEHSHIZ","bundle":"https://pith.science/pith/2YBSQ4HGAA27EPNIYQASEHSHIZ/bundle.json","state":"https://pith.science/pith/2YBSQ4HGAA27EPNIYQASEHSHIZ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/2YBSQ4HGAA27EPNIYQASEHSHIZ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:2YBSQ4HGAA27EPNIYQASEHSHIZ","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":"18eb1ef8bf3e0e66e4ff6c88bdf69dd13e7807e9dead304aa840136636ecc821","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2012-03-09T01:12:54Z","title_canon_sha256":"f7c38f4d8bcea3d40a9d339cdcb4e922babb6df85499d9cb94e54246baab5c81"},"schema_version":"1.0","source":{"id":"1203.1967","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1203.1967","created_at":"2026-05-18T04:00:27Z"},{"alias_kind":"arxiv_version","alias_value":"1203.1967v1","created_at":"2026-05-18T04:00:27Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1203.1967","created_at":"2026-05-18T04:00:27Z"},{"alias_kind":"pith_short_12","alias_value":"2YBSQ4HGAA27","created_at":"2026-05-18T12:26:50Z"},{"alias_kind":"pith_short_16","alias_value":"2YBSQ4HGAA27EPNI","created_at":"2026-05-18T12:26:50Z"},{"alias_kind":"pith_short_8","alias_value":"2YBSQ4HG","created_at":"2026-05-18T12:26:50Z"}],"graph_snapshots":[{"event_id":"sha256:89633f3ab1790280829815da90443a0b9065c28823a5fe07a830bf5554719ffa","target":"graph","created_at":"2026-05-18T04:00:27Z","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 $S = K[x_1,..., x_n]$ be a polynomial ring over a field $K$. Let $I(G) \\subseteq S$ denote the edge ideal of a graph $G$. We show that the $\\ell$th symbolic power $I(G)^{(\\ell)}$ is a Cohen-Macaulay ideal (i.e., $S/I(G)^{(\\ell)}$ is Cohen-Macaulay) for some integer $\\ell \\ge 3$ if and only if $G$ is a disjoint union of finitely many complete graphs. When this is the case, all the symbolic powers $I(G)^{(\\ell)}$ are Cohen-Macaulay ideals. Similarly, we characterize graphs $G$ for which $S/I(G)^{(\\ell)}$ has (FLC).\n  As an application, we show that an edge ideal $I(G)$ is complete intersecti","authors_text":"Giancarlo Rinaldo, Ken-ichi Yoshida, Naoki Terai","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2012-03-09T01:12:54Z","title":"Cohen--Macaulaynees for symbolic power ideals of edge ideals"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1203.1967","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:fedf4b4edb4a63caeaa87704eb4ba25c2c550655f616bddabbf5524184b9b729","target":"record","created_at":"2026-05-18T04:00:27Z","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":"18eb1ef8bf3e0e66e4ff6c88bdf69dd13e7807e9dead304aa840136636ecc821","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2012-03-09T01:12:54Z","title_canon_sha256":"f7c38f4d8bcea3d40a9d339cdcb4e922babb6df85499d9cb94e54246baab5c81"},"schema_version":"1.0","source":{"id":"1203.1967","kind":"arxiv","version":1}},"canonical_sha256":"d6032870e60035f23da8c401221e474654093034a0fcfe81c29566e6c11eb720","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"d6032870e60035f23da8c401221e474654093034a0fcfe81c29566e6c11eb720","first_computed_at":"2026-05-18T04:00:27.880796Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:00:27.880796Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"38j9UOisIyyu01PoJU79oLNpP8blb5SGHCYpwMntXoJ0d4Ln5c2/s4fzHtojt0mlQPJTSQYcynScQ9i6J0TeBA==","signature_status":"signed_v1","signed_at":"2026-05-18T04:00:27.881489Z","signed_message":"canonical_sha256_bytes"},"source_id":"1203.1967","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:fedf4b4edb4a63caeaa87704eb4ba25c2c550655f616bddabbf5524184b9b729","sha256:89633f3ab1790280829815da90443a0b9065c28823a5fe07a830bf5554719ffa"],"state_sha256":"32b93b4ded5870b763c7b76a3dbd7302328576b02b0838667b6f65bd470b1fb1"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"UnmlRwjZVrKa3n9oHXScjuNGKY054gbtYuo9oZkIdk3dDpxFmGAjY2T5OkEQ7aVtpAhmjeaGGb+QipidoAS/Cg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-06T03:01:37.857096Z","bundle_sha256":"5947a6638d7d2e5dc6c315aa679b156898af3150626a4cd3e35fe59bdd3ad7a3"}}