{"paper":{"title":"On Coloring Random Subgraphs of a Fixed Graph","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DS","math.PR"],"primary_cat":"math.CO","authors_text":"Igor Shinkar","submitted_at":"2016-12-13T19:00:55Z","abstract_excerpt":"Given an arbitrary graph $G$ we study the chromatic number of a random subgraph $G_{1/2}$ obtained from $G$ by removing each edge independently with probability $1/2$. Studying $\\chi(G_{1/2})$ has been suggested by Bukh~\\cite{Bukh}, who asked whether $\\mathbb{E}[\\chi(G_{1/2})] \\geq \\Omega( \\chi(G)/\\log(\\chi(G)))$ holds for all graphs $G$. In this paper we show that for any graph $G$ with chromatic number $k = \\chi(G)$ and for all $d \\leq k^{1/3}$ it holds that $\\Pr[\\chi(G_{1/2}) \\leq d] < \\exp \\left(- \\Omega\\left(\\frac{k(k-d^3)}{d^3}\\right)\\right)$. In particular, $\\Pr[G_{1/2} \\text{ is bipart"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1612.04319","kind":"arxiv","version":2},"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"}