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 C. 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 edges defined as follows. Given p,q∈S, pq is an edge of k-DGC(S) provided 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, denoted k-GGC(S), is defined analogously, except that the homothets considered are restricted to be smallest homothets of C with p and q on the 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 these bounds to 7 for squares, 11 for regular hexagons, 12 for regular octagons, and 11 for even-sided regular t-gons (for t≥10). These constitute the first general results on Hamiltonicity for convex shape Delaunay and Gabriel graphs. In addition, we show lower bounds of k=3 and k=6 on the existence of a bottleneck Hamiltonian cycle in the k-order Gabriel graph for squares and hexagons, respectively. Finally, we construct a point set such that for an infinite family of regular polygons Pt, the Delaunay graph DGPt does not contain a Hamiltonian cycle.

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Computational Geometry
School of Computer Science

Bose, P, Cano, P. (Pilar), Saumell, M. (Maria), & Silveira, R.I. (Rodrigo I.). (2020). Hamiltonicity for convex shape Delaunay and Gabriel graphs. Computational Geometry. doi:10.1016/j.comgeo.2020.101629