10-Gabriel graphs are Hamiltonian
TomΓ‘Ε‘ Kaiser, Maria Saumell, Nico Van Cleemput Β· Information Processing Letters Β· 2015
Given a set S of points in the plane, the k-Gabriel graph of S is the geometric graph with vertex set S , where π π , π π β π are connected by an edge if and only if the closed disk having segment Μ Μ Μ Μ Μ Μ Μ Μ Μ Μ π π β’ π π as diameter contains at most k points of π β { π π , π π } . 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.