Given a set P of n points in the plane, the order-k Delaunay graph is a graph with vertex set P and an edge exists between two points p, q ∈ P when there is a circle through p and q with at most k other points of P in its interior. We provide upper and lower bounds on the number of edges in an order-k Delaunay graph. We study the combinatorial structure of the set of triangulations that can be constructed with edges of this graph. Furthermore, we show that the order-k Delaunay graph is connected under the flip operation when k ≤ 1 but not necessarily connected for other values of k. If P is in convex position then the order-k Delaunay graph is connected for all k ≥ 0. We show that the order-k Gabriel graph, a subgraph of the order-k Delaunay graph, is Hamiltonian for k ≥ 15. Finally, the order-k Delaunay graph can be used to efficiently solve a coloring problem with applications to frequency assignments in cellular networks.

We provide a new lower bound on the number of (≤ k)-edges of a set of n points in the plane in general position. We show that for 0 ≤ k ≤ (n − 2)/2 the number of (≤ k)-edges is at leastwhich, for n/3 ≤ k ≤ 0.4864n, improves the previous best lower bound in [12]. As a main consequence, we obtain a new lower bound on the rectilinear crossing number of the complete graph or, in other words, on the minimum number of convex quadrilaterals determined by n points in the plane in general position. We show that the crossing number is at least 41 108which improves the previous bound of 0.37533 n 4 + O(n 3 ) in [12] and approaches the best known upper bound 0.380559 n 4 + (n 3 ) in [4]. The proof is based on a result about the structure of sets attaining the rectilinear crossing number, for which we show that the convex hull is always a triangle.Further implications include improved results for small values of n. We extend the range of known values for the rectilinear crossing number, namely by cr(K 19 ) = 1318 and cr(K 21 ) = 2055. Moreover, we provide improved upper bounds on the maximum number of halving edges a point set can have.

No abstract

No abstract

No abstract