Topological graphs

Abstract

A simple topological graph is a graph drawn in the plane so that its edges are represented by continuous arcs with the property that any two of them meet at most once. We present a novel tool for finding crossing free subgraphs in simple topological graphs. Using this tool, we solve the following two problems. Let G be a complete simple topological graph on n vertices. The three edges induced by any triplet of vertices in G form a simple closed curve. If this curve contains no vertex in its interior (exterior), then we say that the triplet forms an empty triangle. In 1998, Harborth proved that G has at least 2 empty triangles, and he conjectured that the number of empty triangles is at least 2n/3. We settle Harborth's conjecture in the affirmative.We also present a new proof of a result by Suk stating that every complete simple topological graph on n vertices contains Ω(n1/3) pairwise disjoint edges.