A key concept for many graph layout algorithms is planarity, a graph property that allows to draw vertices and edges crossing-free in the plane. Important is the generalization to $k$-planar graphs, which can be drawn in the plane with at most $k > 0$ crossings per edge. One of the basic graph properties that have been explored for those graph classes is the maximum edge density, i.e., the maximum number of edges a $k$-planar graph on $n$ vertices may have. While there are numerous results for the classes of $1$- and $2$-planar graphs, there are few results for increasing $k=3$ or $4$ due to the complex graph structures. We make a first step towards even larger $k>4$ exploring the class of $5$-planar graphs. While our main tool is still a discharging technique, a better understanding of the structure of the denser parts leads to corresponding density bounds in a much simpler way. We first apply a simplified version of our technique to outer $5$-planar graphs and surprisingly observe that the structure of maximally dense (general) $5$-planar graphs differs from the known uniform structure of maximally dense $k$-planar graphs for smaller $1 \leq k \leq 4$. As the central result of this paper, we then show that graphs that admit a simple 5-planar drawing have at most $7(n-2)$ edges, drastically improving the previous best bound of $\approx8.3n$. This even implies a small improvement of the leading constant in the Crossing Lemma $cr(G) \ge c \frac{m^3}{n^2}$ from $c=\frac{1}{27.48}$ to $c=\frac{1}{27.3}$. To demonstrate the potential of our new technique, we also apply it to 4-planar and 6-planar graphs.

翻译：许多图布局算法的核心概念是平面性，即允许在平面上无交叉地绘制顶点和边的图属性。而推广到$k$-平面图（每条边至多允许$k>0$个交叉）具有重要意义。图的基本属性之一是其最大边密度，即$n$个顶点的$k$-平面图可能拥有的最大边数。尽管关于1-平面图和2-平面图已有大量结果，但随着$k$增至3或4，由于复杂的图结构，相关结论较为稀少。我们首次向更大的$k>4$迈进，探索5-平面图类。虽然主要工具仍是放电技术，但通过对稠密部分结构的更深入理解，能以更简捷的方式导出相应的密度上界。我们首先将简化版技术应用于外5-平面图，并意外发现最大稠密（一般）5-平面图的结构与已知的$1 \leq k \leq 4$最大稠密$k$-平面图的均匀结构存在差异。作为本文的核心结果，我们证明了允许简单5-平面绘制的图最多有$7(n-2)$条边，大幅改进了此前约$8.3n$的最佳上界。这甚至使交叉引理$cr(G) \ge c \frac{m^3}{n^2}$中的主导常数$c$从$\frac{1}{27.48}$微调至$\frac{1}{27.3}$。为展示新技术的潜力，我们还将该方法应用于4-平面图和6-平面图。