{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:3NJ5GXBRJDHAREJBL7MIW5FRTJ","short_pith_number":"pith:3NJ5GXBR","canonical_record":{"source":{"id":"1605.06980","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2016-05-23T11:33:25Z","cross_cats_sorted":["math.AG","math.CO"],"title_canon_sha256":"e3e112b72270f9730ca89c7581db0eb1dee8b64e8ba36916c324b779bd57a78e","abstract_canon_sha256":"da4b8a3aa709b4b9e9346d9cd2f74e4115698fe8060a4d4f6046d7203735fefd"},"schema_version":"1.0"},"canonical_sha256":"db53d35c3148ce0891215fd88b74b19a7b2027417ca788c8f31cf1a4483e0503","source":{"kind":"arxiv","id":"1605.06980","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1605.06980","created_at":"2026-05-18T01:14:10Z"},{"alias_kind":"arxiv_version","alias_value":"1605.06980v1","created_at":"2026-05-18T01:14:10Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1605.06980","created_at":"2026-05-18T01:14:10Z"},{"alias_kind":"pith_short_12","alias_value":"3NJ5GXBRJDHA","created_at":"2026-05-18T12:29:55Z"},{"alias_kind":"pith_short_16","alias_value":"3NJ5GXBRJDHAREJB","created_at":"2026-05-18T12:29:55Z"},{"alias_kind":"pith_short_8","alias_value":"3NJ5GXBR","created_at":"2026-05-18T12:29:55Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:3NJ5GXBRJDHAREJBL7MIW5FRTJ","target":"record","payload":{"canonical_record":{"source":{"id":"1605.06980","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2016-05-23T11:33:25Z","cross_cats_sorted":["math.AG","math.CO"],"title_canon_sha256":"e3e112b72270f9730ca89c7581db0eb1dee8b64e8ba36916c324b779bd57a78e","abstract_canon_sha256":"da4b8a3aa709b4b9e9346d9cd2f74e4115698fe8060a4d4f6046d7203735fefd"},"schema_version":"1.0"},"canonical_sha256":"db53d35c3148ce0891215fd88b74b19a7b2027417ca788c8f31cf1a4483e0503","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:14:10.677457Z","signature_b64":"i5yVDnizcacKZEgSDO4EorW+XWAsZyX2KE79bcGoaY71PDyMnIRn+IhLRx2LntfrYkcEchWgZVdRoM7/BkeMDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"db53d35c3148ce0891215fd88b74b19a7b2027417ca788c8f31cf1a4483e0503","last_reissued_at":"2026-05-18T01:14:10.676784Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:14:10.676784Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1605.06980","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":"2026-05-18T01:14:10Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"zRje3GNtLq/1XYe1s+y6AxLinc+iEssh1N/8KrL/DQyUTsOjGVq+YLDA6VmYxbz74P2tgHm8iezNsTaHigdPBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T17:47:08.805403Z"},"content_sha256":"3e60b03feea18a8fec2f304bc6021d6563a97998ab6b49be65eba292900c8dd7","schema_version":"1.0","event_id":"sha256:3e60b03feea18a8fec2f304bc6021d6563a97998ab6b49be65eba292900c8dd7"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:3NJ5GXBRJDHAREJBL7MIW5FRTJ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Bounds on the regularity of toric ideals of graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG","math.CO"],"primary_cat":"math.AC","authors_text":"Adam Van Tuyl, Augustine O'Keefe, Jennifer Biermann","submitted_at":"2016-05-23T11:33:25Z","abstract_excerpt":"Let $G$ be a finite simple graph. We give a lower bound for the Castelnuovo-Mumford regularity of the toric ideal $I_G$ associated to $G$ in terms of the sizes and number of induced complete bipartite graphs in $G$. When $G$ is a chordal bipartite graph, we find an upper bound for the regularity of $I_G$ in terms of the size of the bipartition of $G$. We also give a new proof for the graded Betti numbers of the toric ideal associated to the complete bipartite graph $K_{2,n}$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1605.06980","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":"2026-05-18T01:14:10Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"udzzZTI1ElnfL1spTDBtOBEekOxtH5/xvlCbunT+3npTj1OibE4C45xSUg5XkQKyhw2yI1eHuVkGZ9Iqh26MCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T17:47:08.806091Z"},"content_sha256":"7fd5bd21b83f72b6468efb05a5343f868adc8c1b546cb162c2e8852fa450dcbe","schema_version":"1.0","event_id":"sha256:7fd5bd21b83f72b6468efb05a5343f868adc8c1b546cb162c2e8852fa450dcbe"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/3NJ5GXBRJDHAREJBL7MIW5FRTJ/bundle.json","state_url":"https://pith.science/pith/3NJ5GXBRJDHAREJBL7MIW5FRTJ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/3NJ5GXBRJDHAREJBL7MIW5FRTJ/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":"2026-05-28T17:47:08Z","links":{"resolver":"https://pith.science/pith/3NJ5GXBRJDHAREJBL7MIW5FRTJ","bundle":"https://pith.science/pith/3NJ5GXBRJDHAREJBL7MIW5FRTJ/bundle.json","state":"https://pith.science/pith/3NJ5GXBRJDHAREJBL7MIW5FRTJ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/3NJ5GXBRJDHAREJBL7MIW5FRTJ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:3NJ5GXBRJDHAREJBL7MIW5FRTJ","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":"da4b8a3aa709b4b9e9346d9cd2f74e4115698fe8060a4d4f6046d7203735fefd","cross_cats_sorted":["math.AG","math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2016-05-23T11:33:25Z","title_canon_sha256":"e3e112b72270f9730ca89c7581db0eb1dee8b64e8ba36916c324b779bd57a78e"},"schema_version":"1.0","source":{"id":"1605.06980","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1605.06980","created_at":"2026-05-18T01:14:10Z"},{"alias_kind":"arxiv_version","alias_value":"1605.06980v1","created_at":"2026-05-18T01:14:10Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1605.06980","created_at":"2026-05-18T01:14:10Z"},{"alias_kind":"pith_short_12","alias_value":"3NJ5GXBRJDHA","created_at":"2026-05-18T12:29:55Z"},{"alias_kind":"pith_short_16","alias_value":"3NJ5GXBRJDHAREJB","created_at":"2026-05-18T12:29:55Z"},{"alias_kind":"pith_short_8","alias_value":"3NJ5GXBR","created_at":"2026-05-18T12:29:55Z"}],"graph_snapshots":[{"event_id":"sha256:7fd5bd21b83f72b6468efb05a5343f868adc8c1b546cb162c2e8852fa450dcbe","target":"graph","created_at":"2026-05-18T01:14:10Z","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 $G$ be a finite simple graph. We give a lower bound for the Castelnuovo-Mumford regularity of the toric ideal $I_G$ associated to $G$ in terms of the sizes and number of induced complete bipartite graphs in $G$. When $G$ is a chordal bipartite graph, we find an upper bound for the regularity of $I_G$ in terms of the size of the bipartition of $G$. We also give a new proof for the graded Betti numbers of the toric ideal associated to the complete bipartite graph $K_{2,n}$.","authors_text":"Adam Van Tuyl, Augustine O'Keefe, Jennifer Biermann","cross_cats":["math.AG","math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2016-05-23T11:33:25Z","title":"Bounds on the regularity of toric ideals of graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1605.06980","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":"sha256:3e60b03feea18a8fec2f304bc6021d6563a97998ab6b49be65eba292900c8dd7","target":"record","created_at":"2026-05-18T01:14:10Z","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":"da4b8a3aa709b4b9e9346d9cd2f74e4115698fe8060a4d4f6046d7203735fefd","cross_cats_sorted":["math.AG","math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2016-05-23T11:33:25Z","title_canon_sha256":"e3e112b72270f9730ca89c7581db0eb1dee8b64e8ba36916c324b779bd57a78e"},"schema_version":"1.0","source":{"id":"1605.06980","kind":"arxiv","version":1}},"canonical_sha256":"db53d35c3148ce0891215fd88b74b19a7b2027417ca788c8f31cf1a4483e0503","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"db53d35c3148ce0891215fd88b74b19a7b2027417ca788c8f31cf1a4483e0503","first_computed_at":"2026-05-18T01:14:10.676784Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:14:10.676784Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"i5yVDnizcacKZEgSDO4EorW+XWAsZyX2KE79bcGoaY71PDyMnIRn+IhLRx2LntfrYkcEchWgZVdRoM7/BkeMDw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:14:10.677457Z","signed_message":"canonical_sha256_bytes"},"source_id":"1605.06980","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:3e60b03feea18a8fec2f304bc6021d6563a97998ab6b49be65eba292900c8dd7","sha256:7fd5bd21b83f72b6468efb05a5343f868adc8c1b546cb162c2e8852fa450dcbe"],"state_sha256":"5d5f17d703defe8f39876158bb13f329fd9f3f906840bddb3b66270dc0e1741d"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"B6sZMocPwEM2qJ9ujtb2KmDXmwQWMI2CI5k+4C0p09K4rQrbYBNI9g3Yr/p3cZFVi61nlgLyny8X+eRAjXfNCg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-28T17:47:08.809821Z","bundle_sha256":"df47d48f338c3058508464b60ce324b70820e0aa56d72672033f46bc3f53b0b8"}}