{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:HKWEJJDZKVH6EUUZ6BLAA5LRDZ","short_pith_number":"pith:HKWEJJDZ","canonical_record":{"source":{"id":"1307.1652","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2013-07-05T16:15:33Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"7a6c4362e5a71a942c6e0713d032e076cb24b2e176d5d8c259f5cbf989f27cdc","abstract_canon_sha256":"d9345d25166703fcb6bf44062fb63dc37f5b306145ef7d4d4d75be21cbd2f862"},"schema_version":"1.0"},"canonical_sha256":"3aac44a479554fe25299f0560075711e66c0152f28a63702cfeed05a940ef75f","source":{"kind":"arxiv","id":"1307.1652","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1307.1652","created_at":"2026-05-18T03:19:12Z"},{"alias_kind":"arxiv_version","alias_value":"1307.1652v1","created_at":"2026-05-18T03:19:12Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1307.1652","created_at":"2026-05-18T03:19:12Z"},{"alias_kind":"pith_short_12","alias_value":"HKWEJJDZKVH6","created_at":"2026-05-18T12:27:46Z"},{"alias_kind":"pith_short_16","alias_value":"HKWEJJDZKVH6EUUZ","created_at":"2026-05-18T12:27:46Z"},{"alias_kind":"pith_short_8","alias_value":"HKWEJJDZ","created_at":"2026-05-18T12:27:46Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:HKWEJJDZKVH6EUUZ6BLAA5LRDZ","target":"record","payload":{"canonical_record":{"source":{"id":"1307.1652","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2013-07-05T16:15:33Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"7a6c4362e5a71a942c6e0713d032e076cb24b2e176d5d8c259f5cbf989f27cdc","abstract_canon_sha256":"d9345d25166703fcb6bf44062fb63dc37f5b306145ef7d4d4d75be21cbd2f862"},"schema_version":"1.0"},"canonical_sha256":"3aac44a479554fe25299f0560075711e66c0152f28a63702cfeed05a940ef75f","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:19:12.281358Z","signature_b64":"HUO5CQXIrCQ66t6PXG8CED6t2f3zKGChYZPBsxSvpm75wF2vneXRQq1oA8KFl+ko4yh5OE5OVohr3om1ISjvCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3aac44a479554fe25299f0560075711e66c0152f28a63702cfeed05a940ef75f","last_reissued_at":"2026-05-18T03:19:12.280481Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:19:12.280481Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1307.1652","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":"2026-05-18T03:19:12Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"NSSuBB4wRK4JJQC1jAv3QkfyzyxyCpYn+qrAEzGmzBMuBBz5oCnD8thaw45j8c5i8ykmLFqAP1zwhWkuQXVzDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T04:23:25.817768Z"},"content_sha256":"46de8c295dd24a3205508810dc15438c8c58e04d4724707a117d887cf4acf298","schema_version":"1.0","event_id":"sha256:46de8c295dd24a3205508810dc15438c8c58e04d4724707a117d887cf4acf298"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:HKWEJJDZKVH6EUUZ6BLAA5LRDZ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The arithmetical rank of the edge ideals of graphs with whiskers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.AC","authors_text":"Antonio Macchia","submitted_at":"2013-07-05T16:15:33Z","abstract_excerpt":"We consider the edge ideals of large classes of graphs with whiskers and for these ideals we prove that the arithmetical rank is equal to the big height. Then we extend these results to other classes of squarefree monomial ideals, generated in any degree, proving that the same equality holds."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1307.1652","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":"2026-05-18T03:19:12Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"+8PM4c90svTKUEgdozLc/FXHMwwBKgek/fH1+S0drGBMt72V4CX6hXnGI5TERY8aGm2E5OqAmKDyNMRQQPi5Cw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T04:23:25.818110Z"},"content_sha256":"9423dc54b460a0d1d9900c9b29aeab63f1d26a878b216b3fc34a88f16df3f268","schema_version":"1.0","event_id":"sha256:9423dc54b460a0d1d9900c9b29aeab63f1d26a878b216b3fc34a88f16df3f268"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/HKWEJJDZKVH6EUUZ6BLAA5LRDZ/bundle.json","state_url":"https://pith.science/pith/HKWEJJDZKVH6EUUZ6BLAA5LRDZ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/HKWEJJDZKVH6EUUZ6BLAA5LRDZ/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":"2026-06-23T04:23:25Z","links":{"resolver":"https://pith.science/pith/HKWEJJDZKVH6EUUZ6BLAA5LRDZ","bundle":"https://pith.science/pith/HKWEJJDZKVH6EUUZ6BLAA5LRDZ/bundle.json","state":"https://pith.science/pith/HKWEJJDZKVH6EUUZ6BLAA5LRDZ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/HKWEJJDZKVH6EUUZ6BLAA5LRDZ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:HKWEJJDZKVH6EUUZ6BLAA5LRDZ","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":"d9345d25166703fcb6bf44062fb63dc37f5b306145ef7d4d4d75be21cbd2f862","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2013-07-05T16:15:33Z","title_canon_sha256":"7a6c4362e5a71a942c6e0713d032e076cb24b2e176d5d8c259f5cbf989f27cdc"},"schema_version":"1.0","source":{"id":"1307.1652","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1307.1652","created_at":"2026-05-18T03:19:12Z"},{"alias_kind":"arxiv_version","alias_value":"1307.1652v1","created_at":"2026-05-18T03:19:12Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1307.1652","created_at":"2026-05-18T03:19:12Z"},{"alias_kind":"pith_short_12","alias_value":"HKWEJJDZKVH6","created_at":"2026-05-18T12:27:46Z"},{"alias_kind":"pith_short_16","alias_value":"HKWEJJDZKVH6EUUZ","created_at":"2026-05-18T12:27:46Z"},{"alias_kind":"pith_short_8","alias_value":"HKWEJJDZ","created_at":"2026-05-18T12:27:46Z"}],"graph_snapshots":[{"event_id":"sha256:9423dc54b460a0d1d9900c9b29aeab63f1d26a878b216b3fc34a88f16df3f268","target":"graph","created_at":"2026-05-18T03:19:12Z","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":"We consider the edge ideals of large classes of graphs with whiskers and for these ideals we prove that the arithmetical rank is equal to the big height. Then we extend these results to other classes of squarefree monomial ideals, generated in any degree, proving that the same equality holds.","authors_text":"Antonio Macchia","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2013-07-05T16:15:33Z","title":"The arithmetical rank of the edge ideals of graphs with whiskers"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1307.1652","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":"sha256:46de8c295dd24a3205508810dc15438c8c58e04d4724707a117d887cf4acf298","target":"record","created_at":"2026-05-18T03:19:12Z","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":"d9345d25166703fcb6bf44062fb63dc37f5b306145ef7d4d4d75be21cbd2f862","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2013-07-05T16:15:33Z","title_canon_sha256":"7a6c4362e5a71a942c6e0713d032e076cb24b2e176d5d8c259f5cbf989f27cdc"},"schema_version":"1.0","source":{"id":"1307.1652","kind":"arxiv","version":1}},"canonical_sha256":"3aac44a479554fe25299f0560075711e66c0152f28a63702cfeed05a940ef75f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"3aac44a479554fe25299f0560075711e66c0152f28a63702cfeed05a940ef75f","first_computed_at":"2026-05-18T03:19:12.280481Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:19:12.280481Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"HUO5CQXIrCQ66t6PXG8CED6t2f3zKGChYZPBsxSvpm75wF2vneXRQq1oA8KFl+ko4yh5OE5OVohr3om1ISjvCg==","signature_status":"signed_v1","signed_at":"2026-05-18T03:19:12.281358Z","signed_message":"canonical_sha256_bytes"},"source_id":"1307.1652","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:46de8c295dd24a3205508810dc15438c8c58e04d4724707a117d887cf4acf298","sha256:9423dc54b460a0d1d9900c9b29aeab63f1d26a878b216b3fc34a88f16df3f268"],"state_sha256":"d225a4cf681fad5f6963c678bcf6732d68ceaaf3a4b17d4494b32e2d8f63c1b7"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"NqEQIT8vPJQlrRZy/sO0ZXpbLouk2rDSiensBoBiTT2WEvS0c0DqjGSxYY7rQeuTTsV0oCfV+nBygA9M35tnBQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-23T04:23:25.820012Z","bundle_sha256":"a7c91ed6a634fa96c1e9b5a903df9c3a80f4da8834f5c98b8703ad73cd7fa962"}}