{"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 $G$, $\\chi(H)\\le f(\\omega(H))$. 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"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1809.04278","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"}