Hamiltonian cycles in delaunay complexes

Abstract

We restate a conjecture concerning the existence of Hamiltonian cycles in graphs resulting from the Delaunay triangulation of point sets in the plane R2. We introduce the notion of Delaunay complex, the natural completion of a Delaunay triangulation. We show that Delaunay complexes are necessarily 3-connected. It remains an unsolved problem to prove that Delaunay complexes have Hamiltonian cycles, or to provide a counterexample. On the other hand, using the methods of I. Rivin and W. Thurston to specify a Delaunay triangulation by its list of hyperbolic dihedral angles, we settle a related conjecture. We provide an example to show that the Hamiltonian cycles in a Delaunay complex may not generate all non-degenerate geometric realizations of Delaunay complexes. That is, there are geometric realizations of Delaunay complexes that are not convex sums of Hamiltonian cycles.