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An Ongoing Project to Improve the Rectilinear and the Pseudolinear Crossing Constants
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abstract
A drawing of a graph in the plane is {\it pseudolinear} if the edges of the drawing can be extended to doubly-infinite curves that form an arrangement of pseudolines, that is, any pair of edges crosses precisely once. A special case are {\it rectilinear} drawings where the edges of the graph are drawn as straight line segments. The rectilinear (pseudolinear) crossing number of a graph is the minimum number of pairs of edges of the graph that cross in any of its rectilinear (pseudolinear) drawings. In this paper we describe an ongoing project to continuously obtain better asymptotic upper bounds on the rectilinear and pseudolinear crossing number of the complete graph $K_n$.
Forward citations
Cited by 2 Pith papers
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On the geometric $k$-colored crossing number of $K_n$
Improved asymptotic upper bounds on the geometric k-colored crossing constant for k=2,...,10, such as cr_2 <= 0.11731412 and cr_3 <= 0.06062466.
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On the 2-colored crossing number
For large n, the minimum number of monochromatic crossings in any 2-colored straight-line drawing of K_n is Θ(n^4), between 1/33 and 0.11798016 times C(n,4).
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