its vertex variant [@PW12]. However, in [@BHJ18_counterexamples], this intuition has been disproved in a strong sense: there exist planar graphs $J$ such that $J$-expansions do *not* have the edge-EP property. In particular, this is the case when $J$ is a sufficiently large ladder or a subcubic tree with sufficiently large pathwidth. Finding a characterization for the planar graphs $J$ such that $J$-expansions have the edge-EP property remains an intriguing open problem.\
In this paper, we develop a ball packing technique to establish the edge-EP property of some graph classes. As a warm-up, we first show how it yields a short self-contained proof of the Erdős-Pósa theorem for cycles, along the lines of Raymond and Thilikos [@Raymond2017]; see section \[sec:classicEP\].

After that, we apply the technique to obtain a relatively short proof that long cycles have the edge-EP property, with an improved bounding function.

It has been long known that long cycles (of length at least $\ell$) have the *vertex*-EP property. The first polynomial bounding function is due to Birmelé, Bondy, and Reed [@BBR07], who obtained an $O(k^2\ell)$ bounding function. Then Fiorini and Herinckx [@FH14] improved this to an $O(k\ell \log k)$ bounding function. Using a different approach, this has been 