{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2012:QBLYPCCNF72224FSI4JNKBV4A7","short_pith_number":"pith:QBLYPCCN","schema_version":"1.0","canonical_sha256":"805787884d2ff5ad70b24712d506bc07e1bec79c84e1e0827945ead917bfd7d2","source":{"kind":"arxiv","id":"1206.1376","version":2},"attestation_state":"computed","paper":{"title":"Towards a weighted version of the Hajnal-Szemer\\'edi Theorem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Choongbum Lee, Graeme Kemkes, J\\'ozsef Balogh, Stephen J. Young","submitted_at":"2012-06-07T00:25:27Z","abstract_excerpt":"For a positive integer r>=2, a K_r-factor of a graph is a collection vertex-disjoint copies of K_r which covers all the vertices of the given graph. The celebrated theorem of Hajnal and Szemer\\'edi asserts that every graph on n vertices with minimum degree at least (1-1/r)n contains a K_r-factor. In this note, we propose investigating the relation between minimum degree and existence of perfect K_r-packing for edge-weighted graphs. The main question we study is the following. Suppose that a positive integer r>=2 and a real t in [0,1] is given. What is the minimum weighted degree of K_n that gu"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1206.1376","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-06-07T00:25:27Z","cross_cats_sorted":[],"title_canon_sha256":"f719ea97afb93f2c9ffd9b57fda7bef6eb181110315ca98ad46996e7badf850a","abstract_canon_sha256":"a1267eb2b2292f2041111439aa305f123522116707722e5d971a5dbec4b8ed68"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:27:12.189737Z","signature_b64":"7eraajGWGlIKmwuQCYY9Ox/Y0vPQmWxvhVpplljHQAFB95dhbWL8WIJr19sJCKjqFJ3wOpUbDlA4xCioIe28AQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"805787884d2ff5ad70b24712d506bc07e1bec79c84e1e0827945ead917bfd7d2","last_reissued_at":"2026-05-18T03:27:12.188904Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:27:12.188904Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Towards a weighted version of the Hajnal-Szemer\\'edi Theorem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Choongbum Lee, Graeme Kemkes, J\\'ozsef Balogh, Stephen J. Young","submitted_at":"2012-06-07T00:25:27Z","abstract_excerpt":"For a positive integer r>=2, a K_r-factor of a graph is a collection vertex-disjoint copies of K_r which covers all the vertices of the given graph. The celebrated theorem of Hajnal and Szemer\\'edi asserts that every graph on n vertices with minimum degree at least (1-1/r)n contains a K_r-factor. In this note, we propose investigating the relation between minimum degree and existence of perfect K_r-packing for edge-weighted graphs. The main question we study is the following. Suppose that a positive integer r>=2 and a real t in [0,1] is given. What is the minimum weighted degree of K_n that gu"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1206.1376","kind":"arxiv","version":2},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1206.1376","created_at":"2026-05-18T03:27:12.189048+00:00"},{"alias_kind":"arxiv_version","alias_value":"1206.1376v2","created_at":"2026-05-18T03:27:12.189048+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1206.1376","created_at":"2026-05-18T03:27:12.189048+00:00"},{"alias_kind":"pith_short_12","alias_value":"QBLYPCCNF722","created_at":"2026-05-18T12:27:18.751474+00:00"},{"alias_kind":"pith_short_16","alias_value":"QBLYPCCNF72224FS","created_at":"2026-05-18T12:27:18.751474+00:00"},{"alias_kind":"pith_short_8","alias_value":"QBLYPCCN","created_at":"2026-05-18T12:27:18.751474+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/QBLYPCCNF72224FSI4JNKBV4A7","json":"https://pith.science/pith/QBLYPCCNF72224FSI4JNKBV4A7.json","graph_json":"https://pith.science/api/pith-number/QBLYPCCNF72224FSI4JNKBV4A7/graph.json","events_json":"https://pith.science/api/pith-number/QBLYPCCNF72224FSI4JNKBV4A7/events.json","paper":"https://pith.science/paper/QBLYPCCN"},"agent_actions":{"view_html":"https://pith.science/pith/QBLYPCCNF72224FSI4JNKBV4A7","download_json":"https://pith.science/pith/QBLYPCCNF72224FSI4JNKBV4A7.json","view_paper":"https://pith.science/paper/QBLYPCCN","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1206.1376&json=true","fetch_graph":"https://pith.science/api/pith-number/QBLYPCCNF72224FSI4JNKBV4A7/graph.json","fetch_events":"https://pith.science/api/pith-number/QBLYPCCNF72224FSI4JNKBV4A7/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/QBLYPCCNF72224FSI4JNKBV4A7/action/timestamp_anchor","attest_storage":"https://pith.science/pith/QBLYPCCNF72224FSI4JNKBV4A7/action/storage_attestation","attest_author":"https://pith.science/pith/QBLYPCCNF72224FSI4JNKBV4A7/action/author_attestation","sign_citation":"https://pith.science/pith/QBLYPCCNF72224FSI4JNKBV4A7/action/citation_signature","submit_replication":"https://pith.science/pith/QBLYPCCNF72224FSI4JNKBV4A7/action/replication_record"}},"created_at":"2026-05-18T03:27:12.189048+00:00","updated_at":"2026-05-18T03:27:12.189048+00:00"}