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Wang proved that every directed graph $\\overrightarrow G$ on $3k$ vertices with minimum total degree $\\delta_t(\\overrightarrow G):=\\min_{v\\in V}(deg^-(v)+deg^+(v)) \\ge 3(3k-1)/2$ has a $\\overrightarrow C_3$-factor, where $\\overrightarrow C_3$ is the directed 3-cycle. The degree bound in Wang's result is tight. However, our main result implies that for all integers $"},"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":"1309.4520","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-09-18T01:53:27Z","cross_cats_sorted":[],"title_canon_sha256":"85019d052d253b2f47abf189fd9f350ec729444d8a95a2b9e924e97cd23bd50d","abstract_canon_sha256":"34eb40740c4197e1813dc282dfaf1c1fb9e5485c1e5c4ef6d389ad8012da7967"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:12:59.173566Z","signature_b64":"qsMNHtnJpCe4e/9lBdGPoaL02i4G2zz06SNAulmMaeR7Ypj9R2+oz840W+dFjF4GL0A86xyqmsQwMlm6Ym8FBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"17e8c75a09ad42dd661505b3b3a188329f18f7be43a724cd3faff443ca876578","last_reissued_at":"2026-05-18T03:12:59.172704Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:12:59.172704Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On directed versions of the Corr\\'adi-Hajnal Corollary","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Andrzej Czygrinow, H.A. Kierstead, Theodore Molla","submitted_at":"2013-09-18T01:53:27Z","abstract_excerpt":"For $k \\in \\mathbb N$, Corr\\'adi and Hajnal proved that every graph $G$ on $3k$ vertices with minimum degree $\\delta(G) \\ge 2k$ has a $C_3$-factor, i.e., a partitioning of the vertex set so that each part induces the 3-cycle $C_3$. Wang proved that every directed graph $\\overrightarrow G$ on $3k$ vertices with minimum total degree $\\delta_t(\\overrightarrow G):=\\min_{v\\in V}(deg^-(v)+deg^+(v)) \\ge 3(3k-1)/2$ has a $\\overrightarrow C_3$-factor, where $\\overrightarrow C_3$ is the directed 3-cycle. The degree bound in Wang's result is tight. 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