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For a digraph $D$, define $\\Delta(D) = \\max_{v\\in V(D)} \\sqrt{d^+(v) \\cdot d^-(v)}$.\n  We prove that, for every fixed integer $b\\in\\mathbb{N}$, every digraph $D$ with $\\Delta(D) = \\Delta$ being sufficiently large with respect to $b$ either contains a biclique whose size exceeds $\\Delta-2b$ or has dichromatic number at most $\\Delta-b$.\n  This extends a classical result of Reed to the directe"},"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":"2607.06928","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.CO","submitted_at":"2026-07-08T02:46:55Z","cross_cats_sorted":["cs.DM"],"title_canon_sha256":"526418edf5a0f210b69329eada254d513525c96123c22a362809a1440cf701ee","abstract_canon_sha256":"bdd60be8d84552f3591f7451862f191c12e5a71a4a2c1f2283525da58e06d49e"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-09T00:19:40.272565Z","signature_b64":"lu71ZUA8Kb46LqPqRoKX/z5cIwP44YCLdRAXvxV8XX2Bht/RIRplEetpuJD5hOmlJHPpmMitbjEzgHcVycouDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d2695acc6517cef8fe3020a92107e9f2adf521c37c0a939aed93897c270b1871","last_reissued_at":"2026-07-09T00:19:40.272084Z","signature_status":"signed_v1","first_computed_at":"2026-07-09T00:19:40.272084Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Coloring digraphs with $\\Delta-b$ colors","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"","cross_cats":["cs.DM"],"primary_cat":"math.CO","authors_text":"Ken-ichi Kawarabayashi, Lucas Picasarri-Arrieta","submitted_at":"2026-07-08T02:46:55Z","abstract_excerpt":"The dichromatic number of a digraph is the minimum number of colors needed to partition its vertex set into acyclic subdigraphs. 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