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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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1612.04319","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-12-13T19:00:55Z","cross_cats_sorted":["cs.DS","math.PR"],"title_canon_sha256":"83ff0f1eac1b21c924d873cee80f9159abf8c84cbac88fd7265be4e08edaab7f","abstract_canon_sha256":"f3fec7648131a6d20d6046426bef04850ddaf626e3bdec11e7dacef40590868d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:17:01.999982Z","signature_b64":"vZEdOGZnYh4o50m6a3l3XO7BsaAvXo6VQqXGOZfHac0B3lh6VarsgiOghiKWfQiv4GJ/mQRQLrlEDnztjh/KBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"95f7a53e96a9d730a7b066c24a34ee1782699d8d53a3c281cffa9555bbc29137","last_reissued_at":"2026-05-18T00:17:01.999277Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:17:01.999277Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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 $\\Pr[\\chi(G_{1/2}) \\leq d] < \\exp \\left(- \\Omega\\left(\\frac{k(k-d^3)}{d^3}\\right)\\right)$. 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