{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:RMI3Q7IUQLN5P54LZ67DZO56J6","short_pith_number":"pith:RMI3Q7IU","canonical_record":{"source":{"id":"1207.5553","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2012-07-23T22:37:44Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"a87a5ad574d2b5b7fea0aabb4a377de63985043e4165ea683f30ff958ab7ecf0","abstract_canon_sha256":"44bc0120706b7b1f56173858b268b3df08f7e3b74123b28681480b2dd970d16e"},"schema_version":"1.0"},"canonical_sha256":"8b11b87d1482dbd7f78bcfbe3cbbbe4f9cfd822d1f132339d9b024e16de51947","source":{"kind":"arxiv","id":"1207.5553","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1207.5553","created_at":"2026-05-18T03:50:22Z"},{"alias_kind":"arxiv_version","alias_value":"1207.5553v1","created_at":"2026-05-18T03:50:22Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1207.5553","created_at":"2026-05-18T03:50:22Z"},{"alias_kind":"pith_short_12","alias_value":"RMI3Q7IUQLN5","created_at":"2026-05-18T12:27:20Z"},{"alias_kind":"pith_short_16","alias_value":"RMI3Q7IUQLN5P54L","created_at":"2026-05-18T12:27:20Z"},{"alias_kind":"pith_short_8","alias_value":"RMI3Q7IU","created_at":"2026-05-18T12:27:20Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:RMI3Q7IUQLN5P54LZ67DZO56J6","target":"record","payload":{"canonical_record":{"source":{"id":"1207.5553","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2012-07-23T22:37:44Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"a87a5ad574d2b5b7fea0aabb4a377de63985043e4165ea683f30ff958ab7ecf0","abstract_canon_sha256":"44bc0120706b7b1f56173858b268b3df08f7e3b74123b28681480b2dd970d16e"},"schema_version":"1.0"},"canonical_sha256":"8b11b87d1482dbd7f78bcfbe3cbbbe4f9cfd822d1f132339d9b024e16de51947","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:50:22.577007Z","signature_b64":"xdEvp+hto9FxdH3Nfx71JLvqSdH0cpQkXk0l5fZR+SsYl9Y8CfbPsVnYmqPwd8Oo962GfOO5LPtEFBw3+lwpDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8b11b87d1482dbd7f78bcfbe3cbbbe4f9cfd822d1f132339d9b024e16de51947","last_reissued_at":"2026-05-18T03:50:22.576142Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:50:22.576142Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1207.5553","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":"2026-05-18T03:50:22Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"FU9CZWtEkvJGXYYgrX9XSLQ3UvaMolGfuOF6AIyriqlIvzdLTVU4aGAFoZVUTz6IXaci8rXNUpzw9WxbVBNjDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-12T02:21:33.936397Z"},"content_sha256":"29bd6511b3d1124b7ee5653b6db4b0a37b4e31c4d1e23c2095efda94e1314024","schema_version":"1.0","event_id":"sha256:29bd6511b3d1124b7ee5653b6db4b0a37b4e31c4d1e23c2095efda94e1314024"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:RMI3Q7IUQLN5P54LZ67DZO56J6","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Regularity 3 in edge ideals associated to bipartite graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.AC","authors_text":"Oscar Fern\\'andez-Ramos, Philippe Gimenez","submitted_at":"2012-07-23T22:37:44Z","abstract_excerpt":"We focus in this paper on edge ideals associated to bipartite graphs and give a combinatorial characterization of those having regularity 3. When the regularity is strictly bigger than 3, we determine the first step $i$ in the minimal graded free resolution where there exists a minimal generator of degree $>i+3$, show that at this step the highest degree of a minimal generator is $i+4$, and determine the value of the corresponding graded Betti number $\\beta_{i,i+4}$ in terms of the combinatorics of the associated bipartite graph. The results can then be easily extended to the non-squarefree ca"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1207.5553","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":"2026-05-18T03:50:22Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"H96JCLh4d77VO9SSZPsw/YCNsGuUuaLwNwOYE6u8pWEx2zKH2CKuAnsdGJzrOo0dDi9ptjc7dMxj73VXW0LVDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-12T02:21:33.937138Z"},"content_sha256":"8c137a5843720de9b691db1b6947260d6382e644471e8af27b2c7c250595d413","schema_version":"1.0","event_id":"sha256:8c137a5843720de9b691db1b6947260d6382e644471e8af27b2c7c250595d413"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/RMI3Q7IUQLN5P54LZ67DZO56J6/bundle.json","state_url":"https://pith.science/pith/RMI3Q7IUQLN5P54LZ67DZO56J6/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/RMI3Q7IUQLN5P54LZ67DZO56J6/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":"2026-06-12T02:21:33Z","links":{"resolver":"https://pith.science/pith/RMI3Q7IUQLN5P54LZ67DZO56J6","bundle":"https://pith.science/pith/RMI3Q7IUQLN5P54LZ67DZO56J6/bundle.json","state":"https://pith.science/pith/RMI3Q7IUQLN5P54LZ67DZO56J6/state.json","well_known_bundle":"https://pith.science/.well-known/pith/RMI3Q7IUQLN5P54LZ67DZO56J6/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:RMI3Q7IUQLN5P54LZ67DZO56J6","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":"44bc0120706b7b1f56173858b268b3df08f7e3b74123b28681480b2dd970d16e","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2012-07-23T22:37:44Z","title_canon_sha256":"a87a5ad574d2b5b7fea0aabb4a377de63985043e4165ea683f30ff958ab7ecf0"},"schema_version":"1.0","source":{"id":"1207.5553","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1207.5553","created_at":"2026-05-18T03:50:22Z"},{"alias_kind":"arxiv_version","alias_value":"1207.5553v1","created_at":"2026-05-18T03:50:22Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1207.5553","created_at":"2026-05-18T03:50:22Z"},{"alias_kind":"pith_short_12","alias_value":"RMI3Q7IUQLN5","created_at":"2026-05-18T12:27:20Z"},{"alias_kind":"pith_short_16","alias_value":"RMI3Q7IUQLN5P54L","created_at":"2026-05-18T12:27:20Z"},{"alias_kind":"pith_short_8","alias_value":"RMI3Q7IU","created_at":"2026-05-18T12:27:20Z"}],"graph_snapshots":[{"event_id":"sha256:8c137a5843720de9b691db1b6947260d6382e644471e8af27b2c7c250595d413","target":"graph","created_at":"2026-05-18T03:50:22Z","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 focus in this paper on edge ideals associated to bipartite graphs and give a combinatorial characterization of those having regularity 3. When the regularity is strictly bigger than 3, we determine the first step $i$ in the minimal graded free resolution where there exists a minimal generator of degree $>i+3$, show that at this step the highest degree of a minimal generator is $i+4$, and determine the value of the corresponding graded Betti number $\\beta_{i,i+4}$ in terms of the combinatorics of the associated bipartite graph. The results can then be easily extended to the non-squarefree ca","authors_text":"Oscar Fern\\'andez-Ramos, Philippe Gimenez","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2012-07-23T22:37:44Z","title":"Regularity 3 in edge ideals associated to bipartite graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1207.5553","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":"sha256:29bd6511b3d1124b7ee5653b6db4b0a37b4e31c4d1e23c2095efda94e1314024","target":"record","created_at":"2026-05-18T03:50:22Z","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":"44bc0120706b7b1f56173858b268b3df08f7e3b74123b28681480b2dd970d16e","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2012-07-23T22:37:44Z","title_canon_sha256":"a87a5ad574d2b5b7fea0aabb4a377de63985043e4165ea683f30ff958ab7ecf0"},"schema_version":"1.0","source":{"id":"1207.5553","kind":"arxiv","version":1}},"canonical_sha256":"8b11b87d1482dbd7f78bcfbe3cbbbe4f9cfd822d1f132339d9b024e16de51947","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"8b11b87d1482dbd7f78bcfbe3cbbbe4f9cfd822d1f132339d9b024e16de51947","first_computed_at":"2026-05-18T03:50:22.576142Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:50:22.576142Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"xdEvp+hto9FxdH3Nfx71JLvqSdH0cpQkXk0l5fZR+SsYl9Y8CfbPsVnYmqPwd8Oo962GfOO5LPtEFBw3+lwpDg==","signature_status":"signed_v1","signed_at":"2026-05-18T03:50:22.577007Z","signed_message":"canonical_sha256_bytes"},"source_id":"1207.5553","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:29bd6511b3d1124b7ee5653b6db4b0a37b4e31c4d1e23c2095efda94e1314024","sha256:8c137a5843720de9b691db1b6947260d6382e644471e8af27b2c7c250595d413"],"state_sha256":"5af3ca5c85016338e3873aab424689c6cd3181fb8d8935b8cef5f4387e5a193e"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"4FRMC9Hd78dm+7kpq+4L+4KhfeeXn77iRAjxkNt3Acq/BIn9ZMYsjl0ZkRzO7+O2PucDEYKhSoJ+nPWlPZ2lBg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-12T02:21:33.941619Z","bundle_sha256":"112c90f23189574cc264a0ddb397af7a12788c0e3f5db2b3a4963a0bdb2b129a"}}