{"paper":{"title":"Partitioning $H$-minor free graphs into three subgraphs with no large components","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Chun-Hung Liu, Sang-il Oum","submitted_at":"2015-03-29T00:22:18Z","abstract_excerpt":"We prove that for every graph $H$, if a graph $G$ has no (odd) $H$ minor, then its vertex set $V(G)$ can be partitioned into three sets $X_1$, $X_2$, $X_3$ such that for each~$i$, the subgraph induced on $X_i$ has no component of size larger than a function of~$H$ and the maximum degree of~$G$. This improves a previous result of Alon, Ding, Oporowski and Vertigan~(2003) stating that $V(G)$ can be partitioned into four such sets if $G$ has no $H$ minor. Our theorem generalizes a result of Esperet and Joret~(2014), who proved it for graphs embeddable on a fixed surface and asked whether it is tr"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1503.08371","kind":"arxiv","version":4},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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"}