{"paper":{"title":"Ruling out FPT algorithms for Weighted Coloring on forests","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.CC","math.CO"],"primary_cat":"cs.DS","authors_text":"Ignasi Sau, Julien Baste, J\\'ulio Ara\\'ujo","submitted_at":"2017-03-28T18:01:14Z","abstract_excerpt":"Given a graph $G$, a proper $k$-coloring of $G$ is a partition $c = (S_i)_{i\\in [1,k]}$ of $V(G)$ into $k$ stable sets $S_1,\\ldots, S_{k}$. Given a weight function $w: V(G) \\to \\mathbb{R}^+$, the weight of a color $S_i$ is defined as $w(i) = \\max_{v \\in S_i} w(v)$ and the weight of a coloring $c$ as $w(c) = \\sum_{i=1}^{k}w(i)$. Guan and Zhu [Inf. Process. Lett., 1997] defined the weighted chromatic number of a pair $(G,w)$, denoted by $\\sigma(G,w)$, as the minimum weight of a proper coloring of $G$. For a positive integer $r$, they also defined $\\sigma(G,w;r)$ as the minimum of $w(c)$ among al"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1703.09726","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"}