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The Unit Distance Problem on Spheres
"... For any D > 1 and for any n 2 we construct a set of n points on a sphere in R of diameter D determining at least cn log n unit distances. This improves a previous lower bound of Erdös, Hickerson and Pach (1989). We also construct a set of n points in the plane not containing collinear triples or ..."
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For any D > 1 and for any n 2 we construct a set of n points on a sphere in R of diameter D determining at least cn log n unit distances. This improves a previous lower bound of Erdös, Hickerson and Pach (1989). We also construct a set of n points in the plane not containing collinear triples
The C Unit Distance Graph.
"... We examine some results on coloring the unit distance graph in the plane. In particular, we examine Coulson’s proof that it cannot be 5colored by polygons, and Woodall’s result that Q[i] is 2colorable. The unit distance graph in the plane is the graph whose vertices are the points of C, with edges ..."
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We examine some results on coloring the unit distance graph in the plane. In particular, we examine Coulson’s proof that it cannot be 5colored by polygons, and Woodall’s result that Q[i] is 2colorable. The unit distance graph in the plane is the graph whose vertices are the points of C
Unit distances in three dimensions
 Combin. Probab. Comput
"... We show that the number of unit distances determined by n points in R 3 is O(n 3/2), slightly improving the bound of Clarkson et al. [5], established in 1990. The new proof uses the recently introduced polynomial partitioning technique of Guth and Katz [12]. While this paper was still in a draft sta ..."
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Cited by 15 (4 self)
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We show that the number of unit distances determined by n points in R 3 is O(n 3/2), slightly improving the bound of Clarkson et al. [5], established in 1990. The new proof uses the recently introduced polynomial partitioning technique of Guth and Katz [12]. While this paper was still in a draft
Unit distances and diameters in euclidean spaces
, 2007
"... We show that the maximum number of unit distances or of diameters in a set of n points in ddimensional Euclidean space is attained only by specific types of Lenz constructions, for all d≥4 and n sufficiently large, depending on d. As a corollary we determine the exact maximum number of unit distan ..."
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Cited by 6 (3 self)
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We show that the maximum number of unit distances or of diameters in a set of n points in ddimensional Euclidean space is attained only by specific types of Lenz constructions, for all d≥4 and n sufficiently large, depending on d. As a corollary we determine the exact maximum number of unit
UNIT DISTANCE PROBLEMS
"... Abstract. We study some discrete and continuous variants of the following problem of Erdős: given a finite subset P of R 2 or R 3,whatis the maximum number of pairs (p1,p2) withp1,p2 ∈ P and p1 −p2  =1? 1. ..."
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Abstract. We study some discrete and continuous variants of the following problem of Erdős: given a finite subset P of R 2 or R 3,whatis the maximum number of pairs (p1,p2) withp1,p2 ∈ P and p1 −p2  =1? 1.
On the connectivity of unit distance graphs
 Graphs Combin
, 1996
"... Abstract. For a number eld K R, consider the graph G(Kd), whose vertices are elements of Kd, with an edge between any two points at (Euclidean) distance 1. We show that G(K2) is not connected while G(Kd) is connected for d 5. We also give necessary and sucient conditions for the connectedness of G ..."
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Cited by 1 (0 self)
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Abstract. For a number eld K R, consider the graph G(Kd), whose vertices are elements of Kd, with an edge between any two points at (Euclidean) distance 1. We show that G(K2) is not connected while G(Kd) is connected for d 5. We also give necessary and sucient conditions for the connectedness
ELEVEN UNIT DISTANCE EMBEDDINGS OF THE HEAWOOD
"... Abstract. In this note we present eleven unit distance embeddings of the Heawood graph, i.e. the pointline incidence graph of the finite projective plane of order two, by way of pictures and 15 digit approximations of the coordinates of the vertices. These together with the defining algebraic equat ..."
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Abstract. In this note we present eleven unit distance embeddings of the Heawood graph, i.e. the pointline incidence graph of the finite projective plane of order two, by way of pictures and 15 digit approximations of the coordinates of the vertices. These together with the defining algebraic
Fast regocnition of planar non unit distance graphs
"... Searching the minimum 4regular planar unit distance graph ..."
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Cited by 1 (1 self)
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Searching the minimum 4regular planar unit distance graph
Two notions of unit distance graphs
, 2014
"... A faithful (unit) distance graph in Rd is a graph whose set of vertices is a finite subset of the ddimensional Euclidean space, where two vertices are adjacent if and only if the Euclidean distance between them is exactly 1. A (unit) distance graph in Rd is any subgraph of such a graph. In the fir ..."
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A faithful (unit) distance graph in Rd is a graph whose set of vertices is a finite subset of the ddimensional Euclidean space, where two vertices are adjacent if and only if the Euclidean distance between them is exactly 1. A (unit) distance graph in Rd is any subgraph of such a graph
Results 1  10
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