Intersection Patterns of Edges in Topological Graphs

Abstract

This thesis is devoted to crossing patterns of edges in topological graphs. We consider the following four problems: A thrackle is a graph drawn in the plane such that every pair of edges meet exactly once: either at a common endpoint or in a proper crossing. Conway's Thrackle Conjecture says that a thrackle cannot have more edges than vertices. By a computational approach we improve the previously known upper bound of 1.5n on the maximal number of edges in a thrackle with n vertices to 1.428n. Moreover, our method yields an algorithm with a finite running time that for any e > 0 either verifies the upper bound of (1 + e)n on the maximum number of edges in a thrackle or disproves the conjecture. It is not hard to see that any simple graph admits a poly-line drawing in the plane such that each edge is represented by a polygonal curve with at most three bends, and each edge crossings realizes a prescribed angle α. We show that if we restrict the number of bends per edge to two and allow edges to cross in k different angles, a graph on n vertices admitting such a drawing can have at most O(nk2) edges. This generalizes a previous result of Arikushi et al., in which the authors treated a special case of our problem, where k = 1 and the prescribed angle has 90 degrees. The classical result known as Hanani-Tutte Theorem states that a graph is planar if and only if it admits a drawing in the plane in which each pair of non-adjacent edges crosses an even number of times. We prove the following monotone variant of this result, conjectured by J.Pach and G.Toth. If G has an x-monotone drawing in which every pair of independent edges crosses evenly, then G has an x-monotone embedding (i.e. a drawing without crossings) with the same vertex locations. We show several interesting algorithmic consequences of this result. In a drawing of a graph, two edges form an odd pair if they cross each other an odd number of times. A pair of edges is independent (or non-adjacent) if they share no endpoint. For a graph G we let ocr(G) be the smallest number of odd pairs in a drawing of G and let iocr(G) be the smallest number of independent odd pairs in a drawing of G. We construct a graph G with iocr(G) < ocr(G), answering a question by Szekely, and –for the first time– giving evidence that crossings of adjacent edges may not always be trivial to eliminate.