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

TL;DR

This paper establishes a new, tight lower bound on the uncrossed number of a graph, improving previous bounds and providing constructions that demonstrate asymptotic tightness for dense graphs.

Contribution

It introduces an improved lower bound on the uncrossed number, expressed in terms of the maximum uncrossed subgraph number, and proves its asymptotic tightness for dense graphs.

Findings

New lower bound on uncrossed number for dense graphs

Bound is asymptotically tight for graphs with density around 1/2

Provides constructions matching the lower bound asymptotically

Abstract

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 $\epsilon>0$.