Approximate Min-max Relations for Odd Cycles in Planar Graphs

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We study the ratio between the minimum size of an odd cycle vertex transversal and the maximum size of a collection of vertex-disjoint odd cycles in a planar graph. We show that this ratio is at most 10. For the corresponding edge version of this problem, Král and Voss [7] recently proved that this ratio is at most 2; we also give a short proof of their result.