{"paper":{"title":"New Bounds for Chromatic Polynomials and Chromatic Roots","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Aysel Erey, Jason Brown","submitted_at":"2016-11-29T10:05:57Z","abstract_excerpt":"If $G$ is a $k$-chromatic graph of order $n$ then it is known that the chromatic polynomial of $G$, $\\pi(G,x)$, is at most $x(x-1)\\cdots (x-(k-1))x^{n-k} = (x)_{\\downarrow k}x^{n-k}$ for every $x\\in \\mathbb{N}$. We improve here this bound by showing that \\[ \\pi(G,x) \\leq (x)_{\\downarrow k} (x-1)^{\\Delta(G)-k+1} x^{n-1-\\Delta(G)}\\] for every $x\\in \\mathbb{N},$ where $\\Delta(G)$ is the maximum degree of $G$. Secondly, we show that if $G$ is a connected $k$-chromatic graph of order $n$ where $k\\geq 4$ then $\\pi(G,x)$ is at most $(x)_{\\downarrow k}(x-1)^{n-k}$ for every real $x\\geq n-2+\\left( {n \\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1611.09545","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"}