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We consider right angle crossing (RAC) drawings of graphs in which the edges are represented by polygonal arcs and any two edges can cross only at a right angle. We show that if a graph with n vertices admits a RAC drawing with at most 1 bend or 2 bends per edge, then the number of edges is at most 6.5n and 74.2n, respectively. This is a strengthening of a(More)
Given a set P of n points in R, let d1 > d2 > . . . denote all distinct inter-point distances generated by point pairs in P . It was shown by Schur, Martini, Perles, and Kupitz that there is at most one d-dimensional regular simplex of edge length d1 whose every vertex belongs to P . We extend this result by showing that for any k the number of(More)
We study the impact of metric constraints on the realizability of planar graphs. Let G be a subgraph of a planar graph H (where H is the "host" of G). The graph G is free in H if for every choice of positive lengths for the edges of G, the host H has a planar straight-line embedding that realizes these lengths; and G is extrinsically free in H if all(More)
The inverse degree of a graph is the sum of the reciprocals of the degrees of its vertices. We prove that in any connected planar graph, the diameter is at most 5/2 times the inverse degree, and that this ratio is tight. To develop a crucial surgery method, we begin by proving the simpler related upper bounds (4(|V | − 1) − |E|)/3 and 4|V |2/3|E| on the(More)
Let P be a set of n points in R. It was conjectured by Schur that the maximum number of (d− 1)-dimensional regular simplices of edge length diam(P ), whose every vertex belongs to P , is n. We prove this statement under the condition that any two of the simplices share at least d− 2 vertices and we conjecture that this condition is always satisfied.
In a convex n-gon, let d1 > d2 > · · · denote the set of all distances between pairs of vertices, and let mi be the number of pairs of vertices at distance di from one another. We prove that ∑ i≤k mi ≤ (2k−1)n, making progress towards the conjecture ∑ i≤k mi ≤ kn of Erdős, Lovász, and Vesztergombi. Using a new computational approach, for large enough n, we(More)
In a seminal paper published in 1946, Erdős initiated the investigation of the distribution of distances generated by point sets in metric spaces. In spite of some spectacular partial successes and persistent attacks by generations of mathematicians, most problems raised in Erdős’ paper are still unsolved. Given a set of n points in Rd, let d1 > d2 > d3 > ·(More)