General Strong Bound on the Uncrossed Number via a Tight Bound for the Maximum Uncrossed Subgraph Number

We investigate a very recent concept for visualizing various aspects of a graph in the plane using a collection of drawings introduced by Hlin\v{e}n\'y and Masa\v{r}\'ik [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, Hlin\v{e}n\'y, Masa\v{r}\'ik, 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 $|E(G)|=\epsilon(|V(G)|)^2$ for some $\epsilon>0$): $\lceil\frac{|E(G)|}{c_\epsilon|V(G)|}\rceil \le unc(G)$, where $c_\epsilon\ge 2.82$ is a constant depending on $\epsilon$. 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 $c'_\epsilon=3-\sqrt{(2-\epsilon)}$ in the expression $\lceil\frac{|E(G)|}{c'_\epsilon|V(G)|+o(|V(G)|)}\rceil \le unc(G)$. Hence, it is tight (up to low-order terms) for $\epsilon \approx \frac{1}{2}$ 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 lower-order terms) on dense graphs for all $\epsilon>0$.