The crossing number of a graph $G$ is the minimum number of edge crossings over all drawings of $G$ in the plane. A graph $G$ is $k$-crossing-critical if its crossing number is at least $k$, but if we remove any edge of $G$, its crossing number drops below $k$. There are examples of $k$-crossing-critical graphs that do not have drawings with exactly $k$ crossings. Richter and Thomassen proved in 1993 that if $G$ is $k$-crossing-critical, then its crossing number is at most $2.5k+16$. We improve this bound to $2k+6\sqrt{k}+44$.

中文翻译：

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交叉临界图交叉数的改进

图$G$的交叉数是平面中$G$所有图的最小边交叉数。一个图$G$是$k$-crossing-critical，如果它的交叉数至少是$k$，但是如果我们移除$G$的任何边，它的交叉数会降到$k$以下。有一些 $k$-crossing-critical 图的例子，这些图没有完全 $k$ 交叉的图形。Richter 和 Thomassen 在 1993 年证明，如果 $G$ 是 $k$-crossing-critical，则其交叉数最多为 $2.5k+16$。我们将这个界限提高到 $2k+6\sqrt{k}+44$。