Improved Bounds for the Crossing Numbers of Km, n and Kn

@article{Klerk2006ImprovedBF,
  title={Improved Bounds for the Crossing Numbers of Km, n and Kn},
  author={Etienne de Klerk and John Maharry and Dmitrii V. Pasechnik and R. Bruce Richter and Gelasio Salazar},
  journal={SIAM J. Discret. Math.},
  year={2006},
  volume={20},
  pages={189-202}
}
It has been long conjectured that the crossing number $\Cr(K_{m,n})$ of the complete bipartite graph $K_{m,n}$ equals the Zarankiewicz number $Z(m,n):= \floor{\frac{m-1}{2}} \floor{\frac{m}{2}} \floor{\frac{n-1}{2}} \floor{\frac{n}{2}}$. Another longstanding conjecture states that the crossing number $\Cr(K_n)$ of the complete graph $K_n$ equals $Z(n):=\frac{1}{4}\smallfloor{\frac{n}{2}} \smallfloor{\frac{n-1}{2}} \smallfloor{\frac{n-2}{2}}\smallfloor{\frac{n-3}{2}}$. In this paper we show the… 

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