{"paper":{"title":"Chromatic Number and Dichromatic Polynomial of Digraphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Afrouz Jabalameli, Amir Hossein Ghodrati, Morteza Saghafian, Saeed Akbari","submitted_at":"2017-11-16T19:23:00Z","abstract_excerpt":"Let $G$ be a graph of order $n$. It is well-known that $\\alpha(G)\\geq \\sum_{i=1}^n \\frac{1}{1+d_i}$, where $\\alpha(G)$ is the independence number of $G$ and $d_1,\\ldots,d_n$ is the degree sequence of $G$. We extend this result to digraphs by showing that if $D$ is a digraph with $n$ vertices, then $ \\alpha(D)\\geq \\sum_{i=1}^n \\left( \\frac{1}{1+d_i^+} + \\frac{1}{1+d_i^-}\n  - \\frac{1}{1+d_i}\\right)$, where $\\alpha(D)$ is the maximum size of an acyclic vertex set of $D$. Golowich proved that for any digraph $D$, $\\chi(D)\\leq \\lceil \\frac{4k}{5} \\rceil+2$, where $k=max(\\Delta^+(D),\\Delta^-(D))$. W"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1711.06293","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":""},"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"}