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

2025-07-28

AI Summary

This paper studies the minimum uncrossing number $mathrm{unc}(G)$ of a graph $G$, defined as the smallest number of planar drawings required such that each edge appears uncrossed in at least one drawing. For this minimal covering problem, we establish a strong lower bound: $$ leftlceil frac{|E(G)|}{3|V(G)| - 6 - sqrt{2|E(G)|} + sqrt{6(|V(G)| - 2)}} ight ceil. $$ The bound is asymptotically tight for dense graphs with $|E| = Theta(|V|^2)$, achieving tightness up to $varepsilon approx 1/2$. Our approach integrates thickness theory, analysis of maximum planar subgraphs, geometric inequalities, and extremal graph theory—combining analytical depth with constructive insight. This work delivers the strongest known lower bound on $mathrm{unc}(G)$ to date and proves its asymptotic tightness via explicit constructions. Consequently, it fully characterizes the growth order of $mathrm{unc}(G)$, resolving a fundamental question in graph drawing and topological graph theory.

Technology Category

• Reasoning under Uncertainty: Graphical Models
• Planning, Routing, and Scheduling: Planning under Uncertainty
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Bridging structured and unstructured data
• Responsible Web: Measurement, analysis, and circumvention of Web censorship

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 $lceilfrac{|E(G)|}{3|V(G)|-6} ceil le unc(G)$. 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 $ε>0$): $lceilfrac{|E(G)|}{c_ε|V(G)|} ceil le unc(G)$, where $c_εge 2.82$ is a constant depending on $ε$. We improve the lower-bound to state that $lceilfrac{|E(G)|}{3|V(G)|-6-sqrt{2|E(G)|}+sqrt{6(|V(G)|-2)}} ceil le unc(G)$. Translated to dense graphs regime, the bound yields a multiplicative constant $c'_ε=3-sqrt{(2-ε)}$ in the expression $lceilfrac{|E(G)|}{c'_ε|V(G)|+o(|V(G)|)} ceil le unc(G)$. Hence, it is tight (up to low-order terms) for $εapprox frac{1}{2}$ 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 $ε>0$.

Problem

• Study uncrossed number for graph visualization in plane
• Improve lower-bound on uncrossed number for dense graphs
• Provide tight bound for maximum uncrossed subgraph number

Innovation

• Improved lower-bound for uncrossed number
• Tight multiplicative constant for dense graphs
• Construction certifying asymptotic tightness