General Strong Bound on the Uncrossed Number which is Tight for the Edge Crossing Number

TL;DR

This work establishes a strong, general lower bound on the uncrossed number $unc(G)$ and a matching upper bound on the maximum uncrossed subgraph number $h(G)$, linking these concepts to graph thickness and outerthickness. The main result bounds $h(G)$ by $3n-6 - \sqrt{2m} + \sqrt{6(n-2)}$, which translates, in dense regimes, to tight asymptotics and yields a sharp lower bound on $unc(G)$ via $unc(G) \ge \lceil m / (3n-6-\sqrt{2m}+\sqrt{6(n-2)}) \rceil$. A construction based on wheel graphs demonstrates tightness up to lower-order terms for all densities, while a strengthened bound is derived for triangle-free graphs, together with a triangle-free tightness construction. The paper also posits a tight conjecture for triangle-free graphs, $h(G) \le 2n-4 - \sqrt{m} + \sqrt{2(n-2)}$, inviting future resolution of the density gap and the precise behavior of $unc(G)$ versus $\theta(G)$ and $\theta_o(G)$ in dense regimes.

Abstract

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 $ε>0$.

Figures (6)

• Figure 1: Multiplicative constant $c_\varepsilon$ given by \ref{['thm:old-lb']} (Uncrossed2) depected by dashed blue line and \ref{['eq:dense']} drawn as full red line. Note that our values of $c_\varepsilon$ are tight for $\varepsilon$ tending towards $\frac{1}{2}$ (complete graphs) and towards 0 (planar triangulated graphs).
• Figure 2: The wheel graph on six vertices $W_6$; see \ref{['sec:prelim']} for the formal definition.
• Figure 3: An example of $DG_{5,11}$. Crossed edges are depicted using dotted lines. See the \ref{['constr']} paragraph for the formal description.
• Figure 4: A visual representation of the asymptotical intersection between $\frac{r_3}{n}$ (red line) and $\frac{b_3^{\pm}}{n}$ for $\varepsilon = 0.5$ (blue line). The filled area (in gray) is highlighting the possible values of $\frac{m_i}{n}$. The maximum possible value of $\frac{m_i}{n}$ corresponds to the multiplicative constant $c_\varepsilon$ described in the introduction.
• Figure 5: An example of $DK_{4,4}$. See the \ref{['constr_DKxx']} paragraph for the formal description.

Theorems & Definitions (33)

• Theorem 1.1: Uncrossed2
• Theorem 1.2: Exact $h$ value for complete ringel64 and complete bipartite germanPaper graphs
• Theorem 1.3: Uncrossed Number Lower Bound
• Theorem 1.4: Uncrossed Number Lower Bound for Triangle-free Graphs
• Theorem 1.6: Maximum Uncrossed Subgraph Number Upper Bound
• Theorem 1.7: Maximum Uncrossed Subgraph Tight Construction
• Theorem 1.8: Maximum Uncrossed Subgraph Number Upper Bound for Triangle-Free Graphs
• Theorem 1.9: Maximum Uncrossed Subgraph Construction for Triangle-Free Graphs
• Theorem 2.1: Maximum Uncrossed Subgraph Tight Construction
• Theorem 2.1: ringel64