# Crossing Numbers and Combinatorial Characterization of Monotone Drawings of $$K_n$$Kn

@article{Balko2015CrossingNA,
title={Crossing Numbers and Combinatorial Characterization of Monotone Drawings of \$\$K\_n\$\$Kn},
author={Martin Balko and Radoslav Fulek and Jan Kyn{\vc}l},
journal={Discrete \& Computational Geometry},
year={2015},
volume={53},
pages={107-143}
}
• Published 12 December 2013
• Mathematics
• Discrete & Computational Geometry
In 1958, Hill conjectured that the minimum number of crossings in a drawing of $$K_n$$Kn is exactly $$Z(n) = \frac{1}{4} \big \lfloor \frac{n}{2}\big \rfloor \big \lfloor \frac{n-1}{2}\big \rfloor \big \lfloor \frac{n-2}{2}\big \rfloor \big \lfloor \frac{n-3}{2}\big \rfloor$$Z(n)=14⌊n2⌋⌊n-12⌋⌊n-22⌋⌊n-32⌋. Generalizing the result by Ábrego et al. for 2-page book drawings, we prove this conjecture for plane drawings in which edges are represented by $$x$$x-monotone curves. In fact, our proof…
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