{"paper":{"title":"Dynamic choosability of triangle-free graphs and sparse random graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Jaehoon Kim, Seongmin Ok","submitted_at":"2015-03-16T00:11:34Z","abstract_excerpt":"The \\textit{$r$-dynamic choosability} of a graph $G$, written ${\\rm ch}_r(G)$, is the least $k$ such that whenever each vertex is assigned a list of at least $k$ colors a proper coloring can be chosen from the lists so that every vertex $v$ has at least $\\min\\{d_G(v),r\\}$ neighbors of distinct colors. Let ${\\rm ch}(G)$ denote the choice number of $G$. In this paper, we prove ${\\rm ch}_r(G)\\leq (1+o(1)){\\rm ch}(G)$ when $\\frac{\\Delta(G)}{\\delta(G)}$ is bounded. We also show that there exists a constant $C$ such that for the random graph $G=G(n,p)$ with $\\frac{2}{n}<p\\leq \\frac{1}{2}$, it holds "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1503.04492","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":""},"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"}