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The Erdős–Pósa Property for Long Circuits



For an integer ℓ at least 3, we prove that if G is a graph containing no two vertex-disjoint circuits of length at least ℓ, then there is a set X of at most math formula vertices that intersects all circuits of length at least ℓ. Our result improves the bound math formula due to Birmelé, Bondy, and Reed (The Erdős–Pósa property for long circuits, Combinatorica 27 (2007), 135–145) who conjecture that ℓ vertices always suffice.