{"paper":{"title":"Chromatic number, clique subdivisions, and the conjectures of Haj\\'os and Erd\\H{o}s-Fajtlowicz","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Benny Sudakov, Choongbum Lee, Jacob Fox","submitted_at":"2011-07-11T01:48:12Z","abstract_excerpt":"For a graph $G$, let $\\chi(G)$ denote its chromatic number and $\\sigma(G)$ denote the order of the largest clique subdivision in $G$. Let H(n) be the maximum of $\\chi(G)/\\sigma(G)$ over all $n$-vertex graphs $G$. A famous conjecture of Haj\\'os from 1961 states that $\\sigma(G) \\geq \\chi(G)$ for every graph $G$. That is, $H(n) \\leq 1$ for all positive integers $n$. This conjecture was disproved by Catlin in 1979. Erd\\H{o}s and Fajtlowicz further showed by considering a random graph that $H(n) \\geq cn^{1/2}/\\log n$ for some absolute constant $c>0$. In 1981 they conjectured that this bound is tigh"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1107.1920","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"}