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

A topological graph drawn on a cylinder whose base is horizontal is 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 ∆, 0 < ∆ < n, such that G contains at least max{ n ∆ , c(∆)} pairwise disjoint edges. By combining our result with a result of Tóth we obtain an alternative proof for the best known lower bound of Ω(n 1 3 ) 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.