{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:FPDGP3VEJ67JFMXVRMUONBOCBG","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":"798e741c021ebcc035a1bf01b75a1c06e89519554a13a005e325667c1d85af12","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2012-11-20T01:32:38Z","title_canon_sha256":"a40f24fb987af45bf648dfa3c4382731048a37b760fbba1ce74d4af29a451126"},"schema_version":"1.0","source":{"id":"1211.4647","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1211.4647","created_at":"2026-05-18T03:40:26Z"},{"alias_kind":"arxiv_version","alias_value":"1211.4647v1","created_at":"2026-05-18T03:40:26Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1211.4647","created_at":"2026-05-18T03:40:26Z"},{"alias_kind":"pith_short_12","alias_value":"FPDGP3VEJ67J","created_at":"2026-05-18T12:27:06Z"},{"alias_kind":"pith_short_16","alias_value":"FPDGP3VEJ67JFMXV","created_at":"2026-05-18T12:27:06Z"},{"alias_kind":"pith_short_8","alias_value":"FPDGP3VE","created_at":"2026-05-18T12:27:06Z"}],"graph_snapshots":[{"event_id":"sha256:a007c013e2ae4341c83cb00da78f3fca289569807b4302eec8c86f82f3d43e4f","target":"graph","created_at":"2026-05-18T03:40:26Z","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":"Given a tree T on n vertices, there is an associated ideal I of a polynomial ring in n variables over a field, generated by all paths of a fixed length of T. We show that such an ideal always satisfies the Konig property and classify all trees for which R/I is Cohen-Macaulay. More generally, we show that an ideal I whose generators correspond to any collection of subtrees of T satisfies the Konig property. Since the edge ideal of a simplicial tree has this form, this generalizes a result of Faridi. Moreover, every square-free monomial ideal can be represented (non-uniquely) as a subtree ideal ","authors_text":"Chelsey Paulsen, Daniel Campos, Ryan Gunderson, Susan Morey, Thomas Polstra","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2012-11-20T01:32:38Z","title":"Depths and Cohen-Macaulay Properties of Path Ideals"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1211.4647","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":"sha256:0996354deaab41bac6a0b56ac6b10faaa565ced43f4c013da59ae653c55e5fd6","target":"record","created_at":"2026-05-18T03:40:26Z","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":"798e741c021ebcc035a1bf01b75a1c06e89519554a13a005e325667c1d85af12","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2012-11-20T01:32:38Z","title_canon_sha256":"a40f24fb987af45bf648dfa3c4382731048a37b760fbba1ce74d4af29a451126"},"schema_version":"1.0","source":{"id":"1211.4647","kind":"arxiv","version":1}},"canonical_sha256":"2bc667eea44fbe92b2f58b28e685c2099b1e1e47c4eebe62895ad73accb85a53","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"2bc667eea44fbe92b2f58b28e685c2099b1e1e47c4eebe62895ad73accb85a53","first_computed_at":"2026-05-18T03:40:26.064131Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:40:26.064131Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"f1uVyyXp+ZRrKmkFdclFUO8R/2JhHkEx829ClMuuaqk+tUq+YLfeIoGv6WNFBwbKl+agT/KdCQ1FG/rwZDneBQ==","signature_status":"signed_v1","signed_at":"2026-05-18T03:40:26.064706Z","signed_message":"canonical_sha256_bytes"},"source_id":"1211.4647","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:0996354deaab41bac6a0b56ac6b10faaa565ced43f4c013da59ae653c55e5fd6","sha256:a007c013e2ae4341c83cb00da78f3fca289569807b4302eec8c86f82f3d43e4f"],"state_sha256":"d9ddebd53cf4315b5d2b34cf047d1f9cd05a18c5af4274dfadfe40dac1ce365f"}