Hamiltonicity for convex shape Delaunay and Gabriel graphs

Abstract

We study Hamiltonicity for some of the most general variants of Delaunay and Gabriel graphs. Let S be a point set in the plane. The k-order Delaunay graph of S, denoted k-DGC(S), has vertex set S and edge pq provided that there exists some homothet of 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-GGC(S) is defined analogously, except for the fact that the homothets considered are restricted to be smallest homothets of C with p and q on its boundary. We provide upper bounds on the minimum value of k for which k-GGC(S) is Hamiltonian. Since k-GGC(S) ¿ k-DGC(S), all results carry over to k-DGC(S). In particular, we give upper bounds of 24 for every C and 15 for every point-symmetric 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 = 10).