{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:F5DAZCFZA56QXXZRGR5LEA7ZE6","short_pith_number":"pith:F5DAZCFZ","canonical_record":{"source":{"id":"1302.7076","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2013-02-28T04:18:26Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"eabf498dc3ee140ff9a994cc8f583dd316bd9f86910a6d929b9be18cdb778676","abstract_canon_sha256":"b67310ab901d67667a028c5dcd26c95d7fd216bd06e99f136ee9b2176e2c79eb"},"schema_version":"1.0"},"canonical_sha256":"2f460c88b9077d0bdf31347ab203f927a1b79262b85dc3db2cd8f8609f6dc133","source":{"kind":"arxiv","id":"1302.7076","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1302.7076","created_at":"2026-05-18T03:32:17Z"},{"alias_kind":"arxiv_version","alias_value":"1302.7076v1","created_at":"2026-05-18T03:32:17Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1302.7076","created_at":"2026-05-18T03:32:17Z"},{"alias_kind":"pith_short_12","alias_value":"F5DAZCFZA56Q","created_at":"2026-05-18T12:27:43Z"},{"alias_kind":"pith_short_16","alias_value":"F5DAZCFZA56QXXZR","created_at":"2026-05-18T12:27:43Z"},{"alias_kind":"pith_short_8","alias_value":"F5DAZCFZ","created_at":"2026-05-18T12:27:43Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:F5DAZCFZA56QXXZRGR5LEA7ZE6","target":"record","payload":{"canonical_record":{"source":{"id":"1302.7076","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2013-02-28T04:18:26Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"eabf498dc3ee140ff9a994cc8f583dd316bd9f86910a6d929b9be18cdb778676","abstract_canon_sha256":"b67310ab901d67667a028c5dcd26c95d7fd216bd06e99f136ee9b2176e2c79eb"},"schema_version":"1.0"},"canonical_sha256":"2f460c88b9077d0bdf31347ab203f927a1b79262b85dc3db2cd8f8609f6dc133","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:32:17.575270Z","signature_b64":"hFtfLQbUgXf6w2sOTNL/UUKUMh675aXUHO8PCC1U1VZvdLq2j0ehglyGloURHuwizYgXX92GzJ6lhhNFZunMDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2f460c88b9077d0bdf31347ab203f927a1b79262b85dc3db2cd8f8609f6dc133","last_reissued_at":"2026-05-18T03:32:17.574590Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:32:17.574590Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1302.7076","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":"2026-05-18T03:32:17Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"BaqrWSTrPVFlFXf6EcHKsA3vifiv/rj3CzZt1iRF37+WLgIh+kHIW+Zfax8K3/ijnMWO100IQbo5XcsB4KJQAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T04:50:19.145687Z"},"content_sha256":"8f98d465bcb24eb53e55e3859b2a9f114b48e7d022cbf5afc4bd0268d3c1815f","schema_version":"1.0","event_id":"sha256:8f98d465bcb24eb53e55e3859b2a9f114b48e7d022cbf5afc4bd0268d3c1815f"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:F5DAZCFZA56QXXZRGR5LEA7ZE6","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Combinatorial characterizations of the Cohen-Macaulayness of the second power of edge ideals","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.AC","authors_text":"Do Trong Hoang, Nguyen Cong Minh, Tran Nam Trung","submitted_at":"2013-02-28T04:18:26Z","abstract_excerpt":"Let $I(G)$ be the edge ideal of a simple graph $G$. In this paper, we will give sufficient and necessary combinatorial conditions of $G$ in which the second symbolic and ordinary power of its edge ideal are Cohen-Macaulay (resp. Buchsbaum, generalized Cohen-Macaulay).\n  As an application of our results, we will classify all bipartite graphs in which the second (symbolic) powers are Cohen-Macaulay (resp. Buchsbaum, generalized Cohen-Macaulay)."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1302.7076","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":"2026-05-18T03:32:17Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ABrzdtubnLoJd6yZqWIWtN7J+v3fqHxh4xlLpaZWV0ogOHvxPu5U+4uJ0CCLt9IkVavRJH3/CTfJrmR/G2WnAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T04:50:19.146035Z"},"content_sha256":"f28116948b62018e4fbae2d7b753b3c0f744884220619c7301d97e66789f5741","schema_version":"1.0","event_id":"sha256:f28116948b62018e4fbae2d7b753b3c0f744884220619c7301d97e66789f5741"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/F5DAZCFZA56QXXZRGR5LEA7ZE6/bundle.json","state_url":"https://pith.science/pith/F5DAZCFZA56QXXZRGR5LEA7ZE6/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/F5DAZCFZA56QXXZRGR5LEA7ZE6/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":"2026-06-01T04:50:19Z","links":{"resolver":"https://pith.science/pith/F5DAZCFZA56QXXZRGR5LEA7ZE6","bundle":"https://pith.science/pith/F5DAZCFZA56QXXZRGR5LEA7ZE6/bundle.json","state":"https://pith.science/pith/F5DAZCFZA56QXXZRGR5LEA7ZE6/state.json","well_known_bundle":"https://pith.science/.well-known/pith/F5DAZCFZA56QXXZRGR5LEA7ZE6/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:F5DAZCFZA56QXXZRGR5LEA7ZE6","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":"b67310ab901d67667a028c5dcd26c95d7fd216bd06e99f136ee9b2176e2c79eb","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2013-02-28T04:18:26Z","title_canon_sha256":"eabf498dc3ee140ff9a994cc8f583dd316bd9f86910a6d929b9be18cdb778676"},"schema_version":"1.0","source":{"id":"1302.7076","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1302.7076","created_at":"2026-05-18T03:32:17Z"},{"alias_kind":"arxiv_version","alias_value":"1302.7076v1","created_at":"2026-05-18T03:32:17Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1302.7076","created_at":"2026-05-18T03:32:17Z"},{"alias_kind":"pith_short_12","alias_value":"F5DAZCFZA56Q","created_at":"2026-05-18T12:27:43Z"},{"alias_kind":"pith_short_16","alias_value":"F5DAZCFZA56QXXZR","created_at":"2026-05-18T12:27:43Z"},{"alias_kind":"pith_short_8","alias_value":"F5DAZCFZ","created_at":"2026-05-18T12:27:43Z"}],"graph_snapshots":[{"event_id":"sha256:f28116948b62018e4fbae2d7b753b3c0f744884220619c7301d97e66789f5741","target":"graph","created_at":"2026-05-18T03:32:17Z","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 $I(G)$ be the edge ideal of a simple graph $G$. In this paper, we will give sufficient and necessary combinatorial conditions of $G$ in which the second symbolic and ordinary power of its edge ideal are Cohen-Macaulay (resp. Buchsbaum, generalized Cohen-Macaulay).\n  As an application of our results, we will classify all bipartite graphs in which the second (symbolic) powers are Cohen-Macaulay (resp. Buchsbaum, generalized Cohen-Macaulay).","authors_text":"Do Trong Hoang, Nguyen Cong Minh, Tran Nam Trung","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2013-02-28T04:18:26Z","title":"Combinatorial characterizations of the Cohen-Macaulayness of the second power of edge ideals"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1302.7076","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":"sha256:8f98d465bcb24eb53e55e3859b2a9f114b48e7d022cbf5afc4bd0268d3c1815f","target":"record","created_at":"2026-05-18T03:32:17Z","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":"b67310ab901d67667a028c5dcd26c95d7fd216bd06e99f136ee9b2176e2c79eb","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2013-02-28T04:18:26Z","title_canon_sha256":"eabf498dc3ee140ff9a994cc8f583dd316bd9f86910a6d929b9be18cdb778676"},"schema_version":"1.0","source":{"id":"1302.7076","kind":"arxiv","version":1}},"canonical_sha256":"2f460c88b9077d0bdf31347ab203f927a1b79262b85dc3db2cd8f8609f6dc133","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"2f460c88b9077d0bdf31347ab203f927a1b79262b85dc3db2cd8f8609f6dc133","first_computed_at":"2026-05-18T03:32:17.574590Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:32:17.574590Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"hFtfLQbUgXf6w2sOTNL/UUKUMh675aXUHO8PCC1U1VZvdLq2j0ehglyGloURHuwizYgXX92GzJ6lhhNFZunMDQ==","signature_status":"signed_v1","signed_at":"2026-05-18T03:32:17.575270Z","signed_message":"canonical_sha256_bytes"},"source_id":"1302.7076","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:8f98d465bcb24eb53e55e3859b2a9f114b48e7d022cbf5afc4bd0268d3c1815f","sha256:f28116948b62018e4fbae2d7b753b3c0f744884220619c7301d97e66789f5741"],"state_sha256":"a20a50a8f828fb05831c585ce488fe0ba764b9269cdd9e64c622327daeeb8aa9"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Z0ISfB9aKfm+j8g0b52xvzxYeKzlGuhAsxgTLrYxsvPOr4P8S7KtxyqQbAdKNeMtBxDXdot7dTyeg7KkxzN3CQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-01T04:50:19.148125Z","bundle_sha256":"5cd4fd095c25c0d5da13640eb75e8a0280b24278f1c13d535edc8ab6e5be69ce"}}