  Given a set $P$ of $n$ points in the plane, the order-$k$ Gabriel graph on
$P$, denoted by $k$-$GG$, has an edge between two points $p$ and $q$ if and
only if the closed disk with diameter $pq$ contains at most $k$ points of $P$,
excluding $p$ and $q$. We study matching problems in $k$-$GG$ graphs. We show
that a Euclidean bottleneck perfect matching of $P$ is contained in $10$-$GG$,
but $8$-$GG$ may not have any Euclidean bottleneck perfect matching. In
addition we show that $0$-$GG$ has a matching of size at least $\frac{n-1}{4}$
and this bound is tight. We also prove that $1$-$GG$ has a matching of size at
least $\frac{2(n-1)}{5}$ and $2$-$GG$ has a perfect matching. Finally we
consider the problem of blocking the edges of $k$-$GG$.
