{"paper":{"title":"An optimal chromatic bound for ($P_2+P_3$, gem)-free graphs","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["cs.DM"],"primary_cat":"math.CO","authors_text":"Arnab Char, T. Karthick","submitted_at":"2024-05-28T04:38:40Z","abstract_excerpt":"Given a graph $G$, the parameters $\\chi(G)$ and $\\omega(G)$ respectively denote the chromatic number and the clique number of $G$. A function $f : \\mathbb{N} \\rightarrow \\mathbb{N}$ such that $f(1) = 1$ and $f(x) \\geq x$, for all $x \\in \\mathbb{N}$ is called a $\\chi$-binding function for the given class of graphs $\\cal{G}$ if every $G \\in \\cal{G}$ satisfies $\\chi(G) \\leq f(\\omega(G))$, and the \\emph{smallest $\\chi$-binding function} $f^*$ for $\\cal{G}$ is defined as $f^*(x) := \\max\\{\\chi(G)\\mid G\\in {\\cal G} \\mbox{ and } \\omega(G)=x\\}$. In general, the problem of obtaining the smallest $\\chi$-"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2405.17819","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/2405.17819/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"}