We investigate a very recent concept for visualizing various aspects of a graph in the plane using a collection of drawings introduced by Hliněný and Masařík [GD 2023]. Formally, given a graph $G$, we aim to find an uncrossed collection containing drawings of $G$ in the plane such that each edge of $G$ is not crossed in at least one drawing in the collection. The uncrossed number of $G$ ($unc(G)$) is the smallest integer $k$ such that an uncrossed collection for $G$ of size $k$ exists. The uncrossed number is lower-bounded by the well-known thickness, which is an edge-decomposition of $G$ into planar graphs. This connection gives a trivial lower-bound $\lceil\frac{|E(G)|}{3|V(G)|-6}\rceil \le unc(G)$. In a recent paper, Balko, Hliněný, Masařík, Orthaber, Vogtenhuber, and Wagner [GD 2024] presented the first non-trivial and general lower-bound on the uncrossed number. We summarize it in terms of dense graphs (where $|E(G)|=ε(|V(G)|)^2$ for some $ε>0$): $\lceil\frac{|E(G)|}{c_ε|V(G)|}\rceil \le unc(G)$, where $c_ε\ge 2.82$ is a constant depending on $ε$. We improve the lower-bound to state that $\lceil\frac{|E(G)|}{3|V(G)|-6-\sqrt{2|E(G)|}+\sqrt{6(|V(G)|-2)}}\rceil \le unc(G)$. Translated to dense graphs regime, the bound yields a multiplicative constant $c'_ε=3-\sqrt{(2-ε)}$ in the expression $\lceil\frac{|E(G)|}{c'_ε|V(G)|+o(|V(G)|)}\rceil \le unc(G)$. Hence, it is tight (up to low-order terms) for $ε\approx \frac{1}{2}$ as warranted by complete graphs. In fact, we formulate our result in the language of the maximum uncrossed subgraph number, that is, the maximum number of edges of $G$ that are not crossed in a drawing of $G$ in the plane. In that case, we also provide a construction certifying that our bound is asymptotically tight (up to low-order terms) on dense graphs for all $ε>0$.

翻译：我们研究了一种由Hliněný和Masařík [GD 2023]提出的、使用一组平面绘制来可视化图各方面的新近概念。形式化地，给定图$G$，我们旨在寻找一个包含$G$的平面绘制集合，使得$G$的每条边在该集合中至少存在一幅绘制中不被交叉。图$G$的无交叉数（$unc(G)$）是指存在$G$的无交叉集合的最小整数$k$。无交叉数的下界由著名的厚度（即将$G$分解为平面图的边分解）所界定。这一关联给出了平凡下界$\lceil\frac{|E(G)|}{3|V(G)|-6}\rceil \le unc(G)$。在近期论文中，Balko、Hliněný、Masařík、Orthaber、Vogtenhuber和Wagner [GD 2024]首次提出了无交叉数的非平凡通用下界。我们针对稠密图（其中$|E(G)|=ε(|V(G)|)^2$，$ε>0$）将其概括为：$\lceil\frac{|E(G)|}{c_ε|V(G)|}\rceil \le unc(G)$，其中$c_ε\ge 2.82$为依赖于$ε$的常数。我们改进该下界为：$\lceil\frac{|E(G)|}{3|V(G)|-6-\sqrt{2|E(G)|}+\sqrt{6(|V(G)|-2)}}\rceil \le unc(G)$。转换到稠密图体系，该界在表达式$\lceil\frac{|E(G)|}{c'_ε|V(G)|+o(|V(G)|)}\rceil \le unc(G)$中产生乘法常数$c'_ε=3-\sqrt{(2-ε)}$。因此，对于$ε\approx \frac{1}{2}$的情况（由完全图保证），该界是紧致的（直至低阶项）。实际上，我们采用最大无交叉子图数的语言表述结果，即$G$在平面绘制中不被交叉的最大边数。在此框架下，我们还提供了一个构造，证明对于所有$ε>0$的稠密图，我们的下界在渐近意义下（直至低阶项）是紧致的。