{"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. A biclique is a set of vertices inducing all possible pairs of opposite arcs. 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"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.06928","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2607.06928/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}