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We prove a very general analogue of this result for directed graphs: for any integer $r \\ge 4$ and any sufficiently large multiple $n$ of $r$, if $G$ is a directed graph on $n$ vertices and every vertex is incident to at least $2(1 - 1/r)n - 1$ directed edges, then $G$ can be partitioned into $n/r$ vertex-disjoint subgraphs of size "},"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":"1603.08198","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-03-27T10:19:25Z","cross_cats_sorted":[],"title_canon_sha256":"7ee066a3eb4158dcbec93c8bfdaa2e2c8c0d3df753597853b741c113be3a628d","abstract_canon_sha256":"5ec2e213b0cd06f4d20c5766ffd5088fa3391e8f1b4cbb6b4edac5eab53babe8"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:18:12.953840Z","signature_b64":"GlZKuvrhGSB/DoXRTJCgLW5sFhCuyT1EOTOdH+3BtcekQWZJaKC59pSi/WlmE028zCVA3l6+T9XRKh0MvKl5Ag==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9ed208465216cb1b9a3392b8a0369a0925ff84767f9e9b76e05529de8b4795d2","last_reissued_at":"2026-05-18T01:18:12.953137Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:18:12.953137Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Tiling directed graphs with tournaments","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Andrew Treglown, Andrzej Czygrinow, Louis DeBiasio, Theodore Molla","submitted_at":"2016-03-27T10:19:25Z","abstract_excerpt":"The Hajnal--Szemer\\'edi theorem states that for any integer $r \\ge 1$ and any multiple $n$ of $r$, if $G$ is a graph on $n$ vertices and $\\delta(G) \\ge (1 - 1/r)n$, then $G$ can be partitioned into $n/r$ vertex-disjoint copies of the complete graph on $r$ vertices. We prove a very general analogue of this result for directed graphs: for any integer $r \\ge 4$ and any sufficiently large multiple $n$ of $r$, if $G$ is a directed graph on $n$ vertices and every vertex is incident to at least $2(1 - 1/r)n - 1$ directed edges, then $G$ can be partitioned into $n/r$ vertex-disjoint subgraphs of size "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1603.08198","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1603.08198","created_at":"2026-05-18T01:18:12.953238+00:00"},{"alias_kind":"arxiv_version","alias_value":"1603.08198v1","created_at":"2026-05-18T01:18:12.953238+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1603.08198","created_at":"2026-05-18T01:18:12.953238+00:00"},{"alias_kind":"pith_short_12","alias_value":"T3JAQRSSC3FR","created_at":"2026-05-18T12:30:44.179134+00:00"},{"alias_kind":"pith_short_16","alias_value":"T3JAQRSSC3FRXGRT","created_at":"2026-05-18T12:30:44.179134+00:00"},{"alias_kind":"pith_short_8","alias_value":"T3JAQRSS","created_at":"2026-05-18T12:30:44.179134+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/T3JAQRSSC3FRXGRTSK4KANU2BE","json":"https://pith.science/pith/T3JAQRSSC3FRXGRTSK4KANU2BE.json","graph_json":"https://pith.science/api/pith-number/T3JAQRSSC3FRXGRTSK4KANU2BE/graph.json","events_json":"https://pith.science/api/pith-number/T3JAQRSSC3FRXGRTSK4KANU2BE/events.json","paper":"https://pith.science/paper/T3JAQRSS"},"agent_actions":{"view_html":"https://pith.science/pith/T3JAQRSSC3FRXGRTSK4KANU2BE","download_json":"https://pith.science/pith/T3JAQRSSC3FRXGRTSK4KANU2BE.json","view_paper":"https://pith.science/paper/T3JAQRSS","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1603.08198&json=true","fetch_graph":"https://pith.science/api/pith-number/T3JAQRSSC3FRXGRTSK4KANU2BE/graph.json","fetch_events":"https://pith.science/api/pith-number/T3JAQRSSC3FRXGRTSK4KANU2BE/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/T3JAQRSSC3FRXGRTSK4KANU2BE/action/timestamp_anchor","attest_storage":"https://pith.science/pith/T3JAQRSSC3FRXGRTSK4KANU2BE/action/storage_attestation","attest_author":"https://pith.science/pith/T3JAQRSSC3FRXGRTSK4KANU2BE/action/author_attestation","sign_citation":"https://pith.science/pith/T3JAQRSSC3FRXGRTSK4KANU2BE/action/citation_signature","submit_replication":"https://pith.science/pith/T3JAQRSSC3FRXGRTSK4KANU2BE/action/replication_record"}},"created_at":"2026-05-18T01:18:12.953238+00:00","updated_at":"2026-05-18T01:18:12.953238+00:00"}