{"paper":{"title":"Substitution and $\\chi$-Boundedness","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Alex Scott, Irena Penev, Maria Chudnovsky, Nicolas Trotignon","submitted_at":"2013-02-05T18:25:31Z","abstract_excerpt":"A class $\\mathcal{G}$ of graphs is said to be {\\em $\\chi$-bounded} if there is a function $f:\\mathbb{N} \\rightarrow \\mathbb{R}$ such that for all $G \\in \\mathcal{G}$ and all induced subgraphs $H$ of $G$, $\\chi(H) \\leq f(\\omega(H))$. In this paper, we show that if $\\mathcal{G}$ is a $\\chi$-bounded class, then so is the closure of $\\mathcal{G}$ under any one of the following three operations: substitution, gluing along a clique, and gluing along a bounded number of vertices. Furthermore, if $\\mathcal{G}$ is $\\chi$-bounded by a polynomial (respectively: exponential) function, then the closure of "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1302.1145","kind":"arxiv","version":1},"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"}