{"paper":{"title":"Sparse Partitions of Graphs with Bounded Clique Number","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Ant\\'onio Gir\\~ao, Toby Insley","submitted_at":"2024-11-29T18:25:25Z","abstract_excerpt":"We prove that for each integer $r\\geq 2$, there exists a constant $C_r>0$ with the following property: for any $0<\\varepsilon \\leq 1/2$ and any graph $G$ with clique number at most $r,$ there is a partition of $V(G)$ into at most $(1/\\varepsilon)^{C_r}$ sets $S_1, \\dots, S_t,$ such that $G[S_i]$ has maximum degree at most $\\varepsilon |S_i|$ for each $1 \\leq i \\leq t.$ This answers a question of Fox, Nguyen, Scott and Seymour, who proved a similar result for graphs with no induced $P_4.$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2411.19915","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/2411.19915/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"}