{"paper":{"title":"$(\\Delta-1)$-dicolouring of digraphs","license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","headline":"","cross_cats":["cs.DM"],"primary_cat":"math.CO","authors_text":"Ararat Harutyunyan, Gil Puig i Surroca, Ken-ichi Kawarabayashi, Lucas Picasarri-Arrieta","submitted_at":"2025-07-14T13:37:44Z","abstract_excerpt":"In 1977, Borodin and Kostochka conjectured that every graph with maximum degree $\\Delta \\geq 9$ is $(\\Delta-1)$-colourable, unless it contains a clique of size $\\Delta$. In 1999, Reed confirmed the conjecture when $\\Delta\\geq 10^{14}$.\n  We propose different generalisations of this conjecture for digraphs, and prove the analogue of Reed's result for each of them. The chromatic number and clique number are replaced respectively by the dichromatic number and the biclique number of digraphs. If $D$ is a digraph such that $\\min(\\tilde{\\Delta}(D),\\Delta^+(D)) = \\Delta \\geq 9$, we conjecture that $D"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2507.10266","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/2507.10266/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"}