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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 $(G, \\psi)$. 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