{"paper":{"title":"Partitioning into degenerate graphs in linear time","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DS"],"primary_cat":"math.CO","authors_text":"Alexandre Talon, Carl Feghali, Daniel Gon\\c{c}alves, H\\'el\\`ene Langlois, Quentin Deschamps, Timoth\\'ee Corsini","submitted_at":"2022-04-23T16:28:42Z","abstract_excerpt":"Let $G$ be a connected graph with maximum degree $\\Delta \\geq 3$ distinct from $K_{\\Delta + 1}$. Generalizing Brooks' Theorem, Borodin, Kostochka and Toft proved that if $p_1, \\dots, p_s$ are non-negative integers such that $p_1 + \\dots + p_s \\geq \\Delta - s$, then $G$ admits a vertex partition into parts $A_1, \\dots, A_s$ such that, for $1 \\leq i \\leq s$, $G[A_i]$ is $p_i$-degenerate. Here we show that such a partition can be performed in linear time. This generalizes previous results that treated subcases of a conjecture of Abu-Khzam, Feghali and Heggernes~\\cite{abu2020partitioning}, which o"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2204.11100","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/2204.11100/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"}