Estimating the number of disjoint edges in simple topological graphs via cylindrical drawings

TL;DR

This paper investigates the minimum number of disjoint edges in simple topological graphs drawn on a cylinder, providing new bounds and an alternative proof for existing results using geometric and combinatorial techniques.

Contribution

It introduces a novel bound relating disjoint edges to angularly monotone drawings and offers an alternative proof for the known lower bound on disjoint edges in complete topological graphs.

Findings

Established a bound linking disjoint edges to angularly monotone drawings.

Provided an alternative proof for the  lower bound on disjoint edges.

Connected geometric properties with combinatorial bounds in topological graph theory.

Abstract

A topological graph drawn on a cylinder whose base is horizontal is \emph{angularly monotone} if every vertical line intersects every edge at most once. Let $c(n)$ denote the maximum number $c$ such that every simple angularly monotone drawing of a complete graph on $n$ vertices contains at least $c$ pairwise disjoint edges. We show that for every simple complete topological graph $G$ there exists $\Delta$, $0<\Delta<n$, such that $G$ contains at least $\max \{\frac n\Delta, c(\Delta)\}$ pairwise disjoint edges. By combining our result with a result of T\'oth we obtain an alternative proof for the best known lower bound of $\Omega(n^\frac 13)$ on the maximum number of pairwise disjoint edges in a simple complete topological graph proved by Suk. Our proof is based on a result of Ruiz-Vargas.