{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2008:TSZLBR4TEY7QGIVYZWGFQSTK2K","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":"3468cb676dcabf493e7890742d22d2682cf36c932737c8ab69357097f71f6b11","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2008-06-06T15:39:02Z","title_canon_sha256":"8894646e88ad425017ed6d9f3796b69679ec75791b911c895b1f604bd5a5d5dd"},"schema_version":"1.0","source":{"id":"0806.1192","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0806.1192","created_at":"2026-05-18T00:39:18Z"},{"alias_kind":"arxiv_version","alias_value":"0806.1192v2","created_at":"2026-05-18T00:39:18Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0806.1192","created_at":"2026-05-18T00:39:18Z"},{"alias_kind":"pith_short_12","alias_value":"TSZLBR4TEY7Q","created_at":"2026-05-18T12:25:58Z"},{"alias_kind":"pith_short_16","alias_value":"TSZLBR4TEY7QGIVY","created_at":"2026-05-18T12:25:58Z"},{"alias_kind":"pith_short_8","alias_value":"TSZLBR4T","created_at":"2026-05-18T12:25:58Z"}],"graph_snapshots":[{"event_id":"sha256:51bd0a4fd0124d7b530e21ce48baf89add15b1f2f93d8a0a813676499683896c","target":"graph","created_at":"2026-05-18T00:39:18Z","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 s<t be two fixed positive integers. We study what are the minimum degree conditions for a bipartite graph G, with both color classes of size n=k(s+t), which ensure that G has a K_{s,t}-factor. Exact result for large n is given. Our result extends the work of Zhao, who determined the minimum degree threshold which guarantees that a bipartite graph has a K_{s,s}-factor.","authors_text":"Jan Hladky, Mathias Schacht","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2008-06-06T15:39:02Z","title":"Note on bipartite graph tilings"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0806.1192","kind":"arxiv","version":2},"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":"sha256:6fc84190ea343fb2009a65ac3d83ad9d748e2ecd3410bcbabf11da5db077da6b","target":"record","created_at":"2026-05-18T00:39:18Z","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":"3468cb676dcabf493e7890742d22d2682cf36c932737c8ab69357097f71f6b11","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2008-06-06T15:39:02Z","title_canon_sha256":"8894646e88ad425017ed6d9f3796b69679ec75791b911c895b1f604bd5a5d5dd"},"schema_version":"1.0","source":{"id":"0806.1192","kind":"arxiv","version":2}},"canonical_sha256":"9cb2b0c793263f0322b8cd8c584a6ad2930243504f3cb46f0a697457064adf73","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"9cb2b0c793263f0322b8cd8c584a6ad2930243504f3cb46f0a697457064adf73","first_computed_at":"2026-05-18T00:39:18.879433Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:39:18.879433Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"8Xrpl692Td8/YaFsCID5snX2PVr3JRyjRCI764ITKK/nuLX5pjtJoxsKB1CfrbuG+xYGAKYx+HeMTYun1DRmDA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:39:18.880118Z","signed_message":"canonical_sha256_bytes"},"source_id":"0806.1192","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:6fc84190ea343fb2009a65ac3d83ad9d748e2ecd3410bcbabf11da5db077da6b","sha256:51bd0a4fd0124d7b530e21ce48baf89add15b1f2f93d8a0a813676499683896c"],"state_sha256":"f8f8cc9757826086b006a096b0af15d0c102e6e891948589acf15e61eb698fb1"}