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This result is equivalent to the following result: If $G$ is a balanced bipartite graph of order $2n$ with partite sets $X$ and $Y$, and if $d_{G}(x)+d_{G}(y) \\ge n + 2$ for every two vertices $x \\in X$ and $y \\in Y$ with $xy \\notin E(G)$, then for every perfect matching $M$, $G$ has a $2$-fac"},"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":"1612.08904","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-12-28T14:56:45Z","cross_cats_sorted":[],"title_canon_sha256":"3b2c2e672ba79b5b6ea3d54c99596ab23c2add46c0906e7a6d96c3f9be9c6608","abstract_canon_sha256":"db0a2aa155fe1cb93ba0ae7327b27b66d1e9177925e7225ef083ef13f4e20186"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:38:46.324337Z","signature_b64":"QP7DINnWjoRjnEp1BvtAeBzQDYsKzyhn5ajsAiRPw8X118xspPDy4yeb8/r0RCKjCdfx2WiMkCeBB/ZfEtPZCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"74dc73a66dc74d77e559c8f1bb4aa17b2169016a7c0871e5a570835950333f6a","last_reissued_at":"2026-05-18T00:38:46.323581Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:38:46.323581Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On directed 2-factors in digraphs and 2-factors containing perfect matchings in bipartite graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Shuya Chiba, Tomoki Yamashita","submitted_at":"2016-12-28T14:56:45Z","abstract_excerpt":"In this paper, we give the following result: If $D$ is a digraph of order $n$, and if $d_{D}^{+}(u) + d_{D}^{-}(v) \\ge n$ for every two distinct vertices $u$ and $v$ with $(u, v) \\notin A(D)$, then $D$ has a directed $2$-factor with exactly $k$ directed cycles of length at least $3$, where $n \\ge 12k+3$. 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