  Given a graph and two integers $k$ and $\ell$, Partial Vertex Cover asks for
a set of at most $k$ vertices whose deletion results in a graph with at most
$\ell$ edges. Based on the expansion lemma, we provide a problem kernel with
$(\ell + 2)(k + \ell)$ vertices. We then introduce a new, additive version of
the expansion lemma and show it can be used to prove a kernel with $(\ell +
1)(k + \ell)$ vertices for $\ell \ge 1$.
