{"paper":{"title":"Chromatic profiles of odd cycles","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Xiaoli Yuan, Yuejian Peng, Zilong Yan","submitted_at":"2024-09-05T10:45:07Z","abstract_excerpt":"Erd\\H{o}s and Simonovits asked the following question: For an integer $c\\geq 2$ and a family of non-bipartite graphs $\\mathcal{F}$, what is the infimum of $\\alpha$ such that any $\\mathcal{F}$-free $n$-vertex graph with $n$ large enough and minimum degree at least $\\alpha n$ has chromatic number at most $c$? Denote the infimum as $\\delta_{\\chi}(\\mathcal{F}, c)$. A fundamental result of Erd\\H{o}s, Stone and Simonovits implies that if $3\\le r+1=\\chi(\\mathcal{F})=\\min\\{\\chi (F): F\\in \\mathcal{F}\\}$, then for any $c\\le r-1$, $\\delta_{\\chi}(\\mathcal{F}, c)=1-{1 \\over r}$. So the remaining challenge "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2409.03407","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2409.03407/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}