{"paper":{"title":"On the chromatic number of some ($P_3\\cup P_2$)-free graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Di Wu, Jinfeng Li, Rui Li","submitted_at":"2023-08-29T12:14:06Z","abstract_excerpt":"A hereditary class $\\cal G$ of graphs is {\\em $\\chi$-bounded} if there is a {\\em $\\chi$-binding function}, say $f$, such that $\\chi(G)\\le f(\\omega(G))$ for every $G\\in\\cal G$, where $\\chi(G)(\\omega(G))$ denotes the chromatic (clique) number of $G$. It is known that for every $(P_3\\cup P_2)$-free graph $G$, $\\chi(G)\\le \\frac{1}{6}\\omega(G)(\\omega(G)+1)(\\omega(G)+2)$ \\cite{BA18}, and the class of $(2K_2, 3K_1)$-free graphs does not admit a linear $\\chi$-binding function\\cite{BBS19}. In this paper, we prove that (\\romannumeral 1) $\\chi(G)\\le2\\omega(G)$ if $G$ is ($P_3\\cup P_2$, kite)-free, (\\roma"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2308.15248","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2308.15248/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"}