Let P be a set of n points in Rd. It was conjectured by Schur that the maximum number of (dâˆ’1)-dimensional regular simplices of edge length diam(P ), whose every vertex belongs to P , is n. We proveâ€¦ (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 mostâ€¦ (More)

By a polygonization of a finite point set S in the plane we understand a simple polygon having S as the set of its vertices. Let B and R be sets of blue and red points, respectively, in the planeâ€¦ (More)

In a convex n-gon, let d1 > d2 > Â· Â· Â· 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 âˆ‘ iâ‰¤k miâ€¦ (More)

Let P be a set of n points in R. It was conjectured by Schur that the maximum number of (dâˆ’ 1)-dimensional regular simplices of edge length diam(P ), whose every vertex belongs to P , is n. We proveâ€¦ (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 thatâ€¦ (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 ofâ€¦ (More)