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If true this is best possible, and we prove this for $k = 1,2,3,4,6$ and all $k\\ge 224,539$.\n  More generally, we conjecture that for every integer $k\\ge 1$, and every pair of reals $\\alpha, \\beta> 0$ with $k\\alpha +\\beta>1$, if $G$ is a bipartite digraph with bipartition $(A,B)$, where every vertex in $A$ has out-degree at least $\\beta|B|$, and every"},"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":"1809.08324","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-09-21T21:43:15Z","cross_cats_sorted":[],"title_canon_sha256":"1c77f2b6ee5bc313eea413c2f7b8cdfbf8327b514d388e881ab435bf1e0b09bc","abstract_canon_sha256":"a242f574af9f773e68f59017ea48d53887a19e8014c5bab749279938441bab86"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:39:41.636363Z","signature_b64":"NUNcLVplQw7IS0p7I7b7u9z1B0pvyu9DgXTvx7oRAKFczyKfDDmpHUKAuFRBmR/b79TBRZ6ETus4tCYoaFLvDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e170f90e69beacea85b51566d98af22811d48de930e1b0bd96797b78d41b854a","last_reissued_at":"2026-05-17T23:39:41.635539Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:39:41.635539Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Short directed cycles in bipartite digraphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Paul Seymour, Sophie Spirkl","submitted_at":"2018-09-21T21:43:15Z","abstract_excerpt":"The Caccetta-H\\\"aggkvist conjecture implies that for every integer $k\\ge 1$, if $G$ is a bipartite digraph, with $n$ vertices in each part, and every vertex has out-degree more than $n/(k+1)$, then $G$ has a directed cycle of length at most $2k$. 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