{"paper":{"title":"An analogue of Reed's conjecture for digraphs","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":"2024-07-08T11:14:41Z","abstract_excerpt":"Reed in 1998 conjectured that every graph $G$ satisfies $\\chi(G) \\leq \\lceil \\frac{\\Delta(G)+1+\\omega(G)}{2} \\rceil$. As a partial result, he proved the existence of $\\varepsilon > 0$ for which every graph $G$ satisfies $\\chi(G) \\leq \\lceil (1-\\varepsilon)(\\Delta(G)+1)+\\varepsilon\\omega(G) \\rceil$. We propose an analogue conjecture for digraphs. Given a digraph $D$, we denote by $\\vec{\\chi}(D)$ the dichromatic number of $D$, which is the minimum number of colours needed to partition $D$ into acyclic induced subdigraphs. We let $\\overleftrightarrow{\\omega}(D)$ denote the size of the largest bic"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2407.05827","kind":"arxiv","version":3},"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/2407.05827/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"}