A graph on $n \ge 3$ vertices drawn in the plane such that each edge is crossed at most four times has at most $6(n-2)$ edges -- this result, proven by Ackerman, is outstanding in the literature of beyond-planar graphs with regard to its tightness and the structural complexity of the graph class. We provide a much shorter proof while at the same time relaxing the conditions on the graph and its embedding, i.e., allowing multi-edges and non-simple drawings.

翻译：在平面上绘制的$n \ge 3$个顶点的图中，每条边至多被交叉四次，则该图最多有$6(n-2)$条边——这一由Ackerman证明的结果，在超越平面图文献中因其紧致性和图类结构的复杂性而著称。我们提供了一个更短的证明，同时放宽了对图及其嵌入的条件，即允许多重边和非简单绘制。