10-Gabriel graphs are Hamiltonian

• We prove that the 10-Gabriel graph of any set of points is Hamiltonian.
• We discuss a possible way to further improve upon the previous result.
• We show that there exist sets of points whose 1-Gabriel graph is not 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∈Spi,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}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.