{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:XMNYHBL3RRTRQNAMK7H7FT6KSQ","short_pith_number":"pith:XMNYHBL3","canonical_record":{"source":{"id":"1109.3180","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-09-14T19:43:02Z","cross_cats_sorted":["cs.DM"],"title_canon_sha256":"077c402bc0d239a70bab39d18ca7286e45bd912f96814f838871686cf0082c27","abstract_canon_sha256":"0fea601aa173401a562baa7f47c674a2c377d5977254daf08455d31578d62c8b"},"schema_version":"1.0"},"canonical_sha256":"bb1b83857b8c6718340c57cff2cfca9424e5520e702258378d83b04be83da6a2","source":{"kind":"arxiv","id":"1109.3180","version":4},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1109.3180","created_at":"2026-05-18T03:24:42Z"},{"alias_kind":"arxiv_version","alias_value":"1109.3180v4","created_at":"2026-05-18T03:24:42Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1109.3180","created_at":"2026-05-18T03:24:42Z"},{"alias_kind":"pith_short_12","alias_value":"XMNYHBL3RRTR","created_at":"2026-05-18T12:26:47Z"},{"alias_kind":"pith_short_16","alias_value":"XMNYHBL3RRTRQNAM","created_at":"2026-05-18T12:26:47Z"},{"alias_kind":"pith_short_8","alias_value":"XMNYHBL3","created_at":"2026-05-18T12:26:47Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:XMNYHBL3RRTRQNAMK7H7FT6KSQ","target":"record","payload":{"canonical_record":{"source":{"id":"1109.3180","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-09-14T19:43:02Z","cross_cats_sorted":["cs.DM"],"title_canon_sha256":"077c402bc0d239a70bab39d18ca7286e45bd912f96814f838871686cf0082c27","abstract_canon_sha256":"0fea601aa173401a562baa7f47c674a2c377d5977254daf08455d31578d62c8b"},"schema_version":"1.0"},"canonical_sha256":"bb1b83857b8c6718340c57cff2cfca9424e5520e702258378d83b04be83da6a2","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:24:42.029101Z","signature_b64":"89aXSwZ4oN7ykDGrV8cj7ivuK/rmPfgvG6CXFbF3jLtxxmP8WeCBDx/O6Z+Tb6Sbjfn2ib1l2G64g4l+FjDpDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"bb1b83857b8c6718340c57cff2cfca9424e5520e702258378d83b04be83da6a2","last_reissued_at":"2026-05-18T03:24:42.028422Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:24:42.028422Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1109.3180","source_version":4,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:24:42Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Ls40SB3ABIR1v43EuqNm/aixtQRESjx3sS7KqNsn8cWQ0dNutDIW4yOWmn4CqcimwC2ieCc+XCvKKk1SjImxBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-30T21:17:04.010921Z"},"content_sha256":"c3ee3bb20c6caf3801dda236260f1c9553c15af54bf81a46e1324e4858557dfb","schema_version":"1.0","event_id":"sha256:c3ee3bb20c6caf3801dda236260f1c9553c15af54bf81a46e1324e4858557dfb"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:XMNYHBL3RRTRQNAMK7H7FT6KSQ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Bisections of graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM"],"primary_cat":"math.CO","authors_text":"Benny Sudakov, Choongbum Lee, Po-Shen Loh","submitted_at":"2011-09-14T19:43:02Z","abstract_excerpt":"A bisection of a graph is a bipartition of its vertex set in which the number of vertices in the two parts differ by at most 1, and its size is the number of edges which go across the two parts. In this paper, motivated by several questions and conjectures of Bollob\\'as and Scott, we study maximum bisections of graphs. First, we extend the classical Edwards bound on maximum cuts to bisections. A simple corollary of our result implies that every graph on $n$ vertices and $m$ edges with no isolated vertices, and maximum degree at most $n/3 + 1$, admits a bisection of size at least $m/2 + n/6$. T"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1109.3180","kind":"arxiv","version":4},"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":"2026-05-18T03:24:42Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"hzuwxqHxdHHyuqkzhXGCwvoFRz6TrNhjcokSeB6FJ/ulNHRhO/8BzRrAJpUy+W82I7gsHoA/tnTS5zRVHExwBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-30T21:17:04.011272Z"},"content_sha256":"b1c1614d7c0a9d99c0cc006f170a56f7f775a66dd3151849cf94845af00e5c8c","schema_version":"1.0","event_id":"sha256:b1c1614d7c0a9d99c0cc006f170a56f7f775a66dd3151849cf94845af00e5c8c"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/XMNYHBL3RRTRQNAMK7H7FT6KSQ/bundle.json","state_url":"https://pith.science/pith/XMNYHBL3RRTRQNAMK7H7FT6KSQ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/XMNYHBL3RRTRQNAMK7H7FT6KSQ/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":"2026-05-30T21:17:04Z","links":{"resolver":"https://pith.science/pith/XMNYHBL3RRTRQNAMK7H7FT6KSQ","bundle":"https://pith.science/pith/XMNYHBL3RRTRQNAMK7H7FT6KSQ/bundle.json","state":"https://pith.science/pith/XMNYHBL3RRTRQNAMK7H7FT6KSQ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/XMNYHBL3RRTRQNAMK7H7FT6KSQ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:XMNYHBL3RRTRQNAMK7H7FT6KSQ","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":"0fea601aa173401a562baa7f47c674a2c377d5977254daf08455d31578d62c8b","cross_cats_sorted":["cs.DM"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-09-14T19:43:02Z","title_canon_sha256":"077c402bc0d239a70bab39d18ca7286e45bd912f96814f838871686cf0082c27"},"schema_version":"1.0","source":{"id":"1109.3180","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1109.3180","created_at":"2026-05-18T03:24:42Z"},{"alias_kind":"arxiv_version","alias_value":"1109.3180v4","created_at":"2026-05-18T03:24:42Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1109.3180","created_at":"2026-05-18T03:24:42Z"},{"alias_kind":"pith_short_12","alias_value":"XMNYHBL3RRTR","created_at":"2026-05-18T12:26:47Z"},{"alias_kind":"pith_short_16","alias_value":"XMNYHBL3RRTRQNAM","created_at":"2026-05-18T12:26:47Z"},{"alias_kind":"pith_short_8","alias_value":"XMNYHBL3","created_at":"2026-05-18T12:26:47Z"}],"graph_snapshots":[{"event_id":"sha256:b1c1614d7c0a9d99c0cc006f170a56f7f775a66dd3151849cf94845af00e5c8c","target":"graph","created_at":"2026-05-18T03:24:42Z","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":"A bisection of a graph is a bipartition of its vertex set in which the number of vertices in the two parts differ by at most 1, and its size is the number of edges which go across the two parts. In this paper, motivated by several questions and conjectures of Bollob\\'as and Scott, we study maximum bisections of graphs. First, we extend the classical Edwards bound on maximum cuts to bisections. A simple corollary of our result implies that every graph on $n$ vertices and $m$ edges with no isolated vertices, and maximum degree at most $n/3 + 1$, admits a bisection of size at least $m/2 + n/6$. T","authors_text":"Benny Sudakov, Choongbum Lee, Po-Shen Loh","cross_cats":["cs.DM"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-09-14T19:43:02Z","title":"Bisections of graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1109.3180","kind":"arxiv","version":4},"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:c3ee3bb20c6caf3801dda236260f1c9553c15af54bf81a46e1324e4858557dfb","target":"record","created_at":"2026-05-18T03:24:42Z","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":"0fea601aa173401a562baa7f47c674a2c377d5977254daf08455d31578d62c8b","cross_cats_sorted":["cs.DM"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-09-14T19:43:02Z","title_canon_sha256":"077c402bc0d239a70bab39d18ca7286e45bd912f96814f838871686cf0082c27"},"schema_version":"1.0","source":{"id":"1109.3180","kind":"arxiv","version":4}},"canonical_sha256":"bb1b83857b8c6718340c57cff2cfca9424e5520e702258378d83b04be83da6a2","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"bb1b83857b8c6718340c57cff2cfca9424e5520e702258378d83b04be83da6a2","first_computed_at":"2026-05-18T03:24:42.028422Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:24:42.028422Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"89aXSwZ4oN7ykDGrV8cj7ivuK/rmPfgvG6CXFbF3jLtxxmP8WeCBDx/O6Z+Tb6Sbjfn2ib1l2G64g4l+FjDpDw==","signature_status":"signed_v1","signed_at":"2026-05-18T03:24:42.029101Z","signed_message":"canonical_sha256_bytes"},"source_id":"1109.3180","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:c3ee3bb20c6caf3801dda236260f1c9553c15af54bf81a46e1324e4858557dfb","sha256:b1c1614d7c0a9d99c0cc006f170a56f7f775a66dd3151849cf94845af00e5c8c"],"state_sha256":"adedc8db888b58ee90889ced3643263be0b6943bcca3418f99b0f3be2202a06f"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"s8U1REaApJAyQzoa24Imc0ph9smyRoxFbyA+SqoscTaLCvV9rCUzbQiBrnkKHBpgFFiIbOLqcE3iENO0FRLyDw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-30T21:17:04.013336Z","bundle_sha256":"f6d545acc13bb3bad7e00656a851a4d6f7e8dd1fe3a859b68ce4f9be1059f820"}}