ball to a vertex, thus obtaining a minor of $G$ with sufficiently large minimum degree $\Omega(k^{1+\epsilon })$ after removing parallel edges. This in turn implies that $G$ has a large clique minor (a clique of order at least $k \cdot |V(J)|$) and hence $G$ contains $k$ vertex-disjoint (and thus also edge-disjoint) $J$-expansions, as desired. This concludes the sketch of the general method.\
Finding adequate reduction operations is of course problem dependent (i.e. it depends on $J$). The main challenge with this method seems to be proving that the ball boundaries are uniformly increasing as a function of their radius once the graph is reduced; ideally exponentially in the radius. This is not difficult in the case of $J=C_3$, because then one can deduce that the balls must induce trees where each internal vertex has degree at least $3$, see Section \[sec:classicEP\]. In the case where $J$ is some larger cycle $C_\ell$, the balls do not necessarily induce trees, but it is still possible to reduce to the case where all balls globally possess some tree-like structure, yielding the desired growth behaviour. This step is the bulk of the proof of Theorem \[thm:longcycles\].\
We wish to reiterate that *if* this method 