Saturated simple and 2-simple topological graphs with few edges

TL;DR

This paper constructs saturated simple and 2-simple topological graphs with linear numbers of edges, improving upper bounds and demonstrating the existence of graphs with low-degree vertices.

Contribution

It provides new upper bounds for saturated simple and 2-simple graphs, showing they can have as few as 7n and 14.5n edges respectively, and extends results to k-simple graphs.

Findings

Saturated simple graphs with 7n edges exist.

Saturated 2-simple graphs with 14.5n edges exist.

New upper bounds for k-simple graphs are established.

Abstract

A simple topological graph is a topological graph in which any two edges have at most one common point, which is either their common endpoint or a proper crossing. More generally, in a k-simple topological graph, every pair of edges has at most k common points of this kind. We construct saturated simple and 2-simple graphs with few edges. These are k-simple graphs in which no further edge can be added. We improve the previous upper bounds of Kyn\v{c}l, Pach, Radoi\v{c}i\'c, and T\'oth and show that there are saturated simple graphs on n vertices with only 7n edges and saturated 2-simple graphs on n vertices with 14.5n edges. As a consequence, 14.5n edges is also a new upper bound for k-simple graphs (considering all values of k). We also construct saturated simple and 2-simple graphs that have some vertices with low degree.