Saturated simple and 2-simple topological graphs with few edges

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 Kynai¾?l, Pach, Radoiai¾?ic, and Tothi¾?[Comput.i¾?Geom.,i¾?48, 2015] and show that there are saturated simple graphs oni¾?n vertices with only 7ni¾?edges and saturated 2-simple graphs oni¾?n vertices with 14.5ni¾?edges. As a consequence, 14.5ni¾?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.