Hamiltonicity for convex shape Delaunay and Gabriel graphs

Abstract

We study Hamiltonicity for some of the most general variants of Delaunay and Gabriel graphs. Instead of defining these proximity graphs using circles, we use an arbitrary convex shape \(\mathcal {C}\) . Let S be a point set in the plane. The k-order Delaunay graph of S, denoted k- \({DG}_{\mathcal {C}}(S)\) , has vertex set S and edge pq provided that there exists some homothet of \(\mathcal {C}\) with p and q on its boundary and containing at most k points of S different from p and q. The k-order Gabriel graph k- \({GG}_{\mathcal {C}}(S)\) is defined analogously, except for the fact that the homothets considered are restricted to be smallest homothets of \(\mathcal {C}\) with p and q on its boundary. We provide upper bounds on the minimum value of k for which k- \({GG}_{\mathcal {C}}(S)\) is Hamiltonian. Since k- \({GG}_{\mathcal {C}}(S)\) \(\subseteq \) k- \({DG}_{\mathcal {C}}(S)\) , all results carry over to k- \({DG}_{\mathcal {C}}(S)\) . In particular, we give upper bounds of 24 for every \(\mathcal {C}\) and 15 for every point-symmetric \(\mathcal {C}\) . We also improve the bound to 7 for squares, 11 for regular hexagons, 12 for regular octagons, and 11 for even-sided regular t-gons (for \(t \ge 10)\) . These constitute the first general results on Hamiltonicity for convex shape Delaunay and Gabriel graphs.