The Erdős–Pósa Property For Long Circuits

We show that, for every l, the family $\mathcal{F}_{l}$ of circuits of length at least l satisfies the Erdős–Pósa property, with f(k)=13l(k−1)(k−2)+(2l+3)(k−1), thereby sharpening a result of C. Thomassen. We obtain as a corollary that graphs without k disjoint circuits of length l or more have tree-width O(lk²).