10-Gabriel graphs are Hamiltonian

Given a set S of points in the plane, the k-Gabriel graph of S is the geometric graph with vertex set S, where pi, pj ∈ S are connected by an edge if and only if the closed disk having segment pipj as diameter contains at most k points of S \ {pi, pj}. 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.