Simple topological graphs

This thesis is devoted to the understanding of topological graphs. We consider the following four problems: 1. A \emph{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, at endpoint or at a crossing. 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 \emph{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. 2. A \emph{monotone cylindrical} graph is a topological graph drawn on an open cylinder with an infinite vertical axis satisfying the condition that every vertical line intersects every edge at most once. It is called \emph{simple} if any pair of its edges have at most one point in common: an endpoint or a point at which they properly cross. We say that two edges are \emph{disjoint} if they do not intersect. We show that every simple complete monotone cylindrical graph on $n$ vertices contains $\Omega(n^{1-\epsilon})$ pairwise disjoint edges for any $\epsilon>0$. As a consequence, we show that every simple complete topological graph (drawn in the plane) with $n$ vertices contains $\Omega(n^{\frac 12-\epsilon})$ pairwise disjoint edges for any $\epsilon>0$. By extending some of the ideas here we are then able to get rid of the $\epsilon$ term in the exponent, showing that in fact we can always guarantee a set with $\Omega(n^{\frac 12})$ pairwise disjoint edges. This improves the previous lower bound of $\Omega(n^\frac 13)$ by Suk and independently by Fulek. We remark that our proof implies a polynomial time algorithm for finding this set of pairwise disjoint edges. 3. A {\em tangle} is a graph drawn in the plane such that its edges are represented by continuous arcs, and any two edges share precisely one point, which is either a common endpoint or an interior point at which the two edges are tangent to each other. These points of tangencies are assumed to be distinct. If we drop the last assumption, that is, more than two edges may touch one another at the same point, then the drawing is called a {\em degenerate tangle}. We settle a problem of Pach, Radoi\v ci'c, and T'oth \cite{TTpaper}, by showing that every degenerate tangle has at most as many edges as vertices. We also give a complete characterization of tangles. 4. We show that for a constant $t\in \NN$, every simple topological graph on $n$ vertices has $O(n)$ edges if the graph has no two sets of $t$ edges each such that every edge in one set is disjoint from all edges of the other set (i.e., the complement of the intersection graph of the edges is $K_{t,t}$-free). As an application, we settle the \emph{tangled-thrackle} conjecture formulated by Pach, Radoi\v{c}i'c, and T'oth: Every $n$-vertex graph drawn in the plane such that every pair of edges have precisely one point in common, where this point is either a common endpoint, a crossing, or a point of tangency, has at most $O(n)$ edges.