L. Lovasz, J. Pach, and M. Szegedy, On Conway's thrackle conjecture, Discrete Comput. Geom. 18 (1997), 369-376. Home/Search Document Details and Download
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Selecting Heavily Covered Points by Pseudo-circles, Spheres .. - Smorodinsky, Sharir (2002) (Correct)
....one of the two possible portions of c pq delimited by p and q, which we choose arbitrarily and denote it by pq . We will show that in the above drawing of G, any two edges on four distinct vertices intersect an even number of times. This, combined with the Hanani Tutte s theorem [16] see also [6, 11]) implies that G is planar (and simple) and hence m 3n 6. Assume to the contrary that there are four vertices of G, p 1 ; q 1 ; p 2 ; q 2 , such that the arc p1 q 1 ( 1 for short) and the arc p2 q2 ( 2 for short) intersect an odd number of times. Since C is a family of pseudo circles, ....
L. Lovasz, J. Pach and M. Szegedy, On Conway's thrackle conjecture, Discrete Comput. Geom. 18 (1997), 369-376.
Ramsey Type Results for Pseudo-circles and Spheres Spanned.. - Smorodinsky, Sharir (2002) (Correct)
....one of the two possible portions of c pq delimited by p and q, which we choose arbitrarily and denote it by pq . We will show that in the above drawing of G, any two edges on four distinct vertices intersect an even number of times. This, combined with the Hanani Tutte s theorem [12] see also [4, 8]) implies that G is planar (and simple) and hence m 3n 6. Assume to the contrary that there are four vertices of G, p 1 ; q 1 ; p 2 ; q 2 , such that the arc p 1 q 1 ( 1 for short) and the arc p 2 q 2 ( 2 for short) intersect an odd number of times. Note that by the assumption that C is ....
L. Lovasz, J. Pach and M. Szegedy, On Conway's thrackle conjecture, Discrete Comput. Geom. 18 (1997), 369-376.
On Generalized Geometric Graphs and Pseudolines - Sharir, Smorodinsky (2001) (Correct)
....are otherwise disjoint, or else they intersect in exactly one point where they cross each other. The notion of a thrackle is due to Conway, who conjectured that the number of edges in a thrackle is at most the number of vertices. The study of thrackles has drawn much attention. Two recent papers [18] and [7] obtain linear bounds for the size of a general thrackle, but with constants of proportionality that are greater than 1. The conjecture is known to hold for straight edge thrackles [20] and, in Section 6, we extend the result, and the proof, to the case of graphs whose edges are ....
L. Lovasz, J, Pach and M. Szegedy, On Conway 's thrackle conjecture, Discrete Comput. Geom. 18 (1997), 369-376.
On Generalized Geometric Graphs and Pseudolines - Sharir, Smorodinsky (2001) (Correct)
....pair of edges either have a common endpoint and are otherwise disjoint, or else they intersect in exactly one point where they cross each other. The notion of a thrackle is due to Conway, who conjectured that the number of edges in a thrackle is at most the number of vertices. Two recent papers [16] and [6] obtain linear bounds for the size of a general thrackle, but with constants of proportionality that are greater than 1. The conjecture is known to hold for straight edge thrackles [17] and, in Section 5, we extend the result, and the proof, to the case of graphs whose edges are ....
L. Lovasz, J, Pach and M. Szegedy, On Conway's thrackle conjecture, Discrete Comput. Geom. 18 (1997), 369-376.
Which Crossing Number is It, Anyway? - Courant Self-citation (Pach) (Correct)
....with crossing number 4, whose rectilinear crossing numbers are arbitrarily large. On the other hand, we cannot rule out the possibility that odd cr(G) pair cr(G) cr(G) for every graph G. The only result in this direction is the following remarkable theorem of Hanani and Tutte (see also [LPS97]) Theorem A. Ch34] T70] If a graph G can be drawn in the plane so that any two edges which do not share an endpoint cross an even number of times, then G is planar. For a generalization of this result to other surfaces, see [CN99] In a xed drawing of a graph G, an edge is called even if ....
L. Lovasz, J. Pach, and M. Szegedy, On Conway's thrackle conjecture, Discrete Comput. Geom. 18 (1997), 369-376.
Which Crossing Number is It Anyway? - Pach, Tóth (1998) Self-citation (Pach) (Correct)
....with crossing number 4, whose rectilinear crossing numbers are arbitrarily large. On the other hand, we cannot rule out the possibility that odd cr(G) pair cr(G) cr(G) for every graph G. The only result in this direction is the following remarkable theorem of Hanani and Tutte (see also [LPS97]) Theorem A. Ch34] T70] If a graph G can be drawn in the plane so that any two edges which do not share an endpoint cross an even number of times, then G is planar. In a fixed drawing of a graph G, an edge is called even if it crosses every other edge an even number of times. It follows ....
L. Lov'asz, J. Pach, and M. Szegedy, On Conway's thrackle conjecture, Discrete Comput. Geom. 18 (1997), 369--376.
On the Complexity of the Union of Fat Objects in - The Plane Alon (Correct)
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L. Lovasz, J. Pach and M. Szegedy, On Conway's thrackle conjecture, Proc. 11th ACM Symp. on Computational Geometry, 1995, 147-151.
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