{"paper":{"title":"An Erd\\H{o}s-P\\'osa theorem for cycles and faces of distinct lengths","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["cs.DM"],"primary_cat":"math.CO","authors_text":"Caleb McFarland, J. Pascal Gollin, Kevin Hendrey, Marek Soko{\\l}owski, Maximilian Gorsky, Meike Hatzel, Paul Wollan, Sebastian Wiederrecht, Tony Huynh","submitted_at":"2026-07-08T00:01:19Z","abstract_excerpt":"We show that for every $k \\in \\mathbb{N}$, every graph $G$ contains $k$ vertex-disjoint cycles of different lengths, or there exists a set $X \\subseteq V(G)$ with $|X| \\in \\mathcal{O}(k^6\\mathsf{polylog}(k))$ such that $G-X$ has at most $k-1$ cycle lengths.\n  We also prove analogous results for facial lengths of embedded graphs. Let $G$ be a graph with a closed 2-cell embedding $\\psi$ on a surface $\\Sigma$ of Euler genus $g$, let $c$ be a colouring of the faces $\\mathcal{F}(\\psi)$ of $\\psi$, and let $R(G,\\psi)$ be the radial graph of $(G, \\psi)$. Then there exist $k$ faces $F_1, \\ldots , F_k \\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.06869","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/2607.06869/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"}