refined by Mousset, Noever, and Škorić, and Weissenberger [@MNSW17] to an $O(k \log k + k\ell)$ bounding function, which is best possible.

After this, Bruhn, Heinlein and Joos [@BHJ19] established that cycles of length at least $\ell$ also have the *edge*-EP property. They obtained an $O(k^2 \log k+ k\ell)$ bounding function. Among other ingredients, they used the vertex-EP property, a reduction which roughly costs them a multiplicative factor $k$. In contrast, we do not require this reduction step; our technique is self-contained and can simultaneously yield the vertex and edge-EP property. This is the main reason we are able to prove an $O(k\ell \cdot \log (k\ell))$ bound:

\[thm:longcycles\] For every integer $\ell\geq 2$, the edge-EP property holds for cycles of length at least $\ell$, with bounding function $8k(\ell-1)(\log_2(k\ell)+1)$.

We remark that for fixed $\ell$, an $O(k \log k)$ bound is best possible. On the other hand, if we let $\ell$ vary, then there might still be some room for improvement. It can be shown [@BHJ19] that it is not possible to improve the bound of Theorem \[thm:longcycles\] below $O(k \log k + k\ell)$ just as in the vertex case, but we are not aware of any better lower bound.\
The 