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 prove that for every centrally symmetric convex polygon Q, there exists a constant α such that any αk-fold covering of the plane by translates of Q can be decomposed into k coverings. This improves on a quadratic upper bound proved by Pach and Tóth (SoCG'07). The question is motivated by a sensor network problem, in which a region has to be monitored by sensors with limited battery lifetime.

Given a tree T on n vertices and a set P of n points in the plane in general position, it is known that T can be straight line embedded in P without crossings. Now imagine the set P is partitioned into two disjoint subsets R and B, and we ask for an embedding of T in P without crossings and with the property that all edges join a point in R (red) and a point in B (blue). In this case we say that T admits a bipartite embedding with respect to the bipartition (R, B). Examples show that the problem in its full generality is not solvable. In view of this fact we consider several embedding problems and study for which bipartitions they can be solved. We present several results that are valid for any bipartition (R, B) in general position, and some other results that hold for particular configurations of points.

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.

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