The classical Crossing Lemma by Ajtai et al.~and Leighton from 1982 gave an important lower bound of $c \frac{m^3}{n^2}$ for the number of crossings in any drawing of a given graph of $n$ vertices and $m$ edges. The original value was $c= 1/100$, which then has gradually been improved. Here, the bounds for the density of $k$-planar graphs played a central role. Our new insight is that for $k=2,3$ the $k$-planar graphs have substantially fewer edges if specific local configurations that occur in drawings of $k$-planar graphs of maximum density are forbidden. Therefore, we are able to derive better bounds for the crossing number $\text{cr}(G)$ of a given graph $G$. In particular, we achieve a bound of $\text{cr}(G) \ge \frac{37}{9}m-\frac{155}{9}(n-2)$ for the range of $5n < m \le 6n$, while our second bound $\text{cr}(G) \ge 5m - \frac{203}{9}(n-2)$ is even stronger for larger $m>6n$.
For $m > 6.77n$, we finally apply the standard probabilistic proof from the BOOK and obtain an improved constant of $c>1/27.48$ in the Crossing Lemma. Note that the previous constant was $1/29$. Although this improvement is not too impressive, we consider our technique as an important new tool, which might be helpful in various other applications.
Université Gustave Eiffel（古斯塔夫·埃菲尔大学）；CNRS, LIGM（法国国家科学研究中心，信息学与机器学习实验室）；Univ Paris Est Creteil, CNRS, LAMA UMR8050（巴黎东克雷泰伊大学，法国国家科学研究中心，LAMA联合研究实验室）；Massachusetts Institute of Technology（麻省理工学院）
Convex drawings of the complete graph: topology meets geometry
本文引入并发展了球面上完全图$K_n$的凸图理论。$K_n$的绘图$D$是凸的，如果对于$K_n$的每个3-循环$T$，存在由$D[T]$界定的闭合圆盘$Δ_T$，使得对于$D[u]$和$D[v]$都在$Δ_T$中的任何两个顶点$u,v$，整个边$D[uv]$也包含在$Δ_T$中。作为这个角度的一个应用程序，我们认为图纸包含一个非凸$K_5$，其扩展到图纸的$K_7$的限制。对于每一个这样的绘图，我们使用凸性来产生一个具有更少交叉的新绘图。这是局部考虑为次优性提供充分条件的第一个例子。特别地，我们不将交叉点的数量{与任何已知图中的交叉点的数量}进行比较。这一结果揭示了Aichholzer的计算机证明（个人通信）表明，对于$n\le 12$，每一个最佳的图纸$K_n$是凸的。凸面图纸的特点是不包括$K_5$五张图纸中的两张。凸图的两种精化是h-凸图和f-凸图。后者已经由Aichholzer等人（Deciding monotonicity of good drawings of the完全图，Proc.~ XVI西班牙计算几何会议（EGC 2015），2015），并独立地，当前文章的作者（列维引理，$K_n$的伪线性绘图，和空三角形，\rbr{J. Graph Theory DOI: 10.1002/jgt.22167)}，等同于伪线性绘图。此外，h-凸图等价于伪球面图，如Arroyo…
