  Given a set $S$ of points in the plane, the $k$-Gabriel graph of $S$ is the
geometric graph with vertex set $S$, where $p_i,p_j\in S$ are connected by an
edge if and only if the closed disk having segment $\bar{p_ip_j}$ as diameter
contains at most $k$ points of $S \setminus \{p_i,p_j\}$. We consider the
following question: What is the minimum value of $k$ such that the $k$-Gabriel
graph of every point set $S$ contains a Hamiltonian cycle? For this value, we
give an upper bound of 10 and a lower bound of 2. The best previously known
values were 15 and 1, respectively.
