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20
Crossing numbers and hard Erdős problems in discrete geometry
 COMBINATORICS, PROBABILITY AND COMPUTING
, 1997
"... We show that an old but not wellknown lower bound for the crossing number of a graph yields short proofs for a number of bounds in discrete plane geometry which were considered hard before: the number of incidences among points and lines, the maximum number of unit distances among n points, the min ..."
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Cited by 107 (1 self)
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We show that an old but not wellknown lower bound for the crossing number of a graph yields short proofs for a number of bounds in discrete plane geometry which were considered hard before: the number of incidences among points and lines, the maximum number of unit distances among n points, the minimum number of distinct distances among n points.
Toward the Rectilinear Crossing Number of K_n : New Embeddings, Upper Bounds, and Asymptotics
, 2000
"... Scheinerman and Wilf [SW94] assert that "an important open problem in the study of graph embeddings is to determine the rectilinear crossing number of the complete graph Kn ." A rectilinear embedding or drawing of Kn is an arrangement of n vertices in the plane, every pair of which is connected by ..."
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Cited by 21 (1 self)
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Scheinerman and Wilf [SW94] assert that "an important open problem in the study of graph embeddings is to determine the rectilinear crossing number of the complete graph Kn ." A rectilinear embedding or drawing of Kn is an arrangement of n vertices in the plane, every pair of which is connected by an edge that is a line segment. We assume that no three vertices are collinear. The rectilinear crossing number of Kn is the fewest number of edge crossings attainable over all planar rectilinear embeddings of Kn . For each n we construct a rectilinear embedding of Kn that has the fewest number of edge crossings and the best asymptotics known to date. Moreover, we give some alternative infinite families of embeddings of Kn with good asymptotics. Finally, we mention some old and new open problems.
The Rectilinear Crossing Number of K_10 is 62
 Electron. J. Combin., 8(1):Research Paper
, 2000
"... The rectilinear crossing number of a graph G is the minimum number of edge crossings that can occur in any drawing of G in which the edges are straight line segments and no three vertices are collinear. This number has been known for G = K n if n # 9. Using a combinatorial argument we show that fo ..."
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Cited by 11 (0 self)
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The rectilinear crossing number of a graph G is the minimum number of edge crossings that can occur in any drawing of G in which the edges are straight line segments and no three vertices are collinear. This number has been known for G = K n if n # 9. Using a combinatorial argument we show that for n =10the number is 62.
On conway’s thrackle conjecture
 Proc. 11th ACM Symp. on Computational Geometry
, 1995
"... A thrackle is a graph drawn in the plane so that its edges are represented by Jordan arcs and any two distinct arcs either meet at exactly one common vertex or cross at exactly one point interior to both arcs. About forty years ago, J. H. Conway conjectured that the number of edges of a thrackle can ..."
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Cited by 11 (2 self)
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A thrackle is a graph drawn in the plane so that its edges are represented by Jordan arcs and any two distinct arcs either meet at exactly one common vertex or cross at exactly one point interior to both arcs. About forty years ago, J. H. Conway conjectured that the number of edges of a thrackle cannot exceed the number of its vertices. We show that a thrackle has at most twice as many edges as vertices. Some related problems and generalizations are also considered. 1
Approximating the Crossing Number of Graphs Embeddable In Any Orientable Surface
"... The crossing number of a graph is the least number of pairwise edge crossings in a drawing of the graph in the plane. We provide an O(n log n) time constant factor approximation algorithm for the crossing number of a graph of bounded maximum degree which is “densely enough” embeddable in an arbitrar ..."
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Cited by 5 (3 self)
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The crossing number of a graph is the least number of pairwise edge crossings in a drawing of the graph in the plane. We provide an O(n log n) time constant factor approximation algorithm for the crossing number of a graph of bounded maximum degree which is “densely enough” embeddable in an arbitrary fixed orientable surface. Our approach combines some known tools with a powerful new lower bound on the crossing number of an embedded graph. This result extends previous results that gave such approximations in particular cases of projective, toroidal or apex graphs; it is a qualitative improvement over previously published algorithms that constructed lowcrossingnumber drawings of embeddable graphs without giving any approximation guarantees. No constant factor approximation algorithms for the crossing number problem over comparably rich classes of graphs are known to date.
Toward the Rectilinear Crossing Number of K n : New Embeddings, Upper Bounds, and Asymptotics
, 2000
"... Scheinerman and Wilf [SW94] assert that "an important open problem in the study of graph embeddings is to determine the rectilinear crossing number of the complete graph Kn ." A rectilinear embedding or drawing of Kn is an arrangement of n vertices in the plane, every pair of which is connected by ..."
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Cited by 3 (1 self)
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Scheinerman and Wilf [SW94] assert that "an important open problem in the study of graph embeddings is to determine the rectilinear crossing number of the complete graph Kn ." A rectilinear embedding or drawing of Kn is an arrangement of n vertices in the plane, every pair of which is connected by an edge that is a line segment. We assume that no three vertices are collinear. The rectilinear crossing number of Kn is the fewest number of edge crossings attainable over all planar rectilinear embeddings of Kn . For each n we construct a rectilinear embedding of Kn that has the fewest number of edge crossings and the best asymptotics known to date. Moreover, we give some alternative infinite families of embeddings of Kn with good asymptotics. Finally, we mention some old and new open problems. 1 Introduction and History Given an arbitrary graph G, determining an embedding of G in the plane that produces the fewest number of edge crossings is NPComplete [GJ83]. The same can be said for a...
On the crossing numbers of complete graphs
 University of Waterloo
, 2006
"... I hereby declare that I am the sole author of this thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I understand that my thesis may be made electronically available to the public. ii In this thesis we prove two main results. The Triangle ..."
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Cited by 2 (1 self)
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I hereby declare that I am the sole author of this thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I understand that my thesis may be made electronically available to the public. ii In this thesis we prove two main results. The Triangle Conjecture asserts that the convex hull of any optimal rectilinear drawing of Kn must be a triangle (for n � 3). We prove that, for the larger class of pseudolinear drawings, the outer face must be a triangle. The other main result is the next step toward Guy’s Conjecture that the crossing number of Kn is 1