G\'eza T\'oth

Crossing numbers of crossing-critical graphs

A graph $G$ is $k$-crossing-critical if $\mbox{cr}(G)\ge k$, but for
any edge $e$ of  $G$, $\mbox{cr}(G-e)< k$.
In 1993 Richter and Thomassen conjectured  that for any
 $k$-crossing-critical graph $G$, $\mbox{cr}(G)\le k+c\sqrt{k}$
 and proved that  $\mbox{cr}(G)\le 5k/2+16$.
 We improve it to $\mbox{cr}(G)\le 2k+6\sqrt{k}+47$
 and review some related results.

Joint work with J\'anos Bar\'at.