{"paper":{"title":"A tight Erd\\H{o}s-P\\'osa function for long cycles","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Andreas Noever, Felix Weissenberger, Frank Mousset, Nemanja \\v{S}kori\\'c","submitted_at":"2016-03-24T14:22:44Z","abstract_excerpt":"A classic result of Erd\\H{o}s and P\\'osa says that any graph contains either $k$ vertex-disjoint cycles or can be made acyclic by deleting at most $O(k \\log k)$ vertices. Here we generalize this result by showing that for all numbers $k$ and $l$ and for every graph $G$, either $G$ contains $k$ vertex-disjoint cycles of length at least $l$, or there exists a set $X$ of $\\mathcal O(kl+k\\log k)$ vertices that meets all cycles of length at least $l$ in $G$. As a corollary, the tree-width of any graph $G$ that does not contain $k$ vertex-disjoint cycles of length at least $l$ is of order $\\mathcal "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1603.07588","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":""},"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"}