Toward the Rectilinear Crossing Number of K_n : New Embeddings, Upper Bounds, and Asymptotics (2000)
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BibTeX
@MISC{Brodsky00towardthe,
author = {Alex Brodsky and Stephane Durocher and Ellen Gethner},
title = {Toward the Rectilinear Crossing Number of K_n : New Embeddings, Upper Bounds, and Asymptotics},
year = {2000}
}
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Abstract
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.
