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)
Let P be a set of n points in R d. 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.
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)
Given a set P of n points in R d , let d 1 > d 2 >. .. 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 d 1 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)
In a seminal paper published in 1946, Erd˝ os 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 Erd˝ os' paper are still unsolved. Given a set of n points in R d , let d 1 > d 2… (More)
In a convex n-gon, let d 1 > d 2 > · · · denote the set of all distances between pairs of vertices, and let m i be the number of pairs of vertices at distance d i from one another. Erd˝ os, Lovász, and Vesztergombi conjectured that i≤k m i ≤ kn. Using a new computational approach, we prove their conjecture when k ≤ 4 and n is large; we also make some… (More)