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In addition, we say that $\\mathcal G$ is polynomially $\\chi$-bounded if $f$ can be taken as a polynomial function. We prove that for every integer $n\\ge3$, there exists a polynomial $f$ such that $\\chi(G)\\le f(\\omega(G))$ for all graphs with no vertex-minor isomorphic to the cycle graph $C_n$. To prove this, we show that if $\\mathcal G$ is polynomially $\\chi$-bounded, then so is the closure of $\\mathcal G$ unde"},"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":"1809.04278","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-09-12T07:04:53Z","cross_cats_sorted":["cs.DM"],"title_canon_sha256":"0cbd89a13327edec52b203e07c93ba34edea00da918f677c95a779a2a026704c","abstract_canon_sha256":"58521329dd1612a6d9ef42f0c32c29417ac1f8586f727615b130e1ddfd4d096e"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:43:23.675205Z","signature_b64":"k203UsMhestqsaSegRcbzwOwBsvFtt4jcItyG+NSuJpPJ2wwpakUV2u7VKpNKoAaUsLA26RNev//QwwMKPvCCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0e808d768427e9d49235268a7c3bfaf4855a536a6143fe10f28bfbbd91cd86be","last_reissued_at":"2026-05-17T23:43:23.674448Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:43:23.674448Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Classes of graphs with no long cycle as a vertex-minor are polynomially $\\chi$-bounded","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM"],"primary_cat":"math.CO","authors_text":"O-joung Kwon, Ringi Kim, Sang-il Oum, Vaidy Sivaraman","submitted_at":"2018-09-12T07:04:53Z","abstract_excerpt":"A class $\\mathcal G$ of graphs is $\\chi$-bounded if there is a function $f$ such that for every graph $G\\in \\mathcal G$ and every induced subgraph $H$ of $G$, $\\chi(H)\\le f(\\omega(H))$. 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