Conway’s Conjecture for Monotone Thrackles

Abstract

A drawing of a graph in the plane is called a thrackle if every pair of edges meet precisely once, either at a common vertex or at a proper crossing. According to Conway's conjecture, every thrackle has at most as many edges as vertices. We prove this conjecture for x-monotone thrackles, that is, in the case when every edge meets every vertical line in at most one point.

On Tangencies Among Planar Curves with an Application to Coloring L-Shapes

“…It is not hard to construct such a family of n (x-monotone) 1-intersecting curves with Ω(n 4/3 ) tangencies based on a famous construction of Erdős (see [10]) of n lines and n points admitting that many point-line incidences. János Pach [9] conjectured that requiring every pair of curves to intersect (either at crossing or a tangency point) leads to significantly less tangencies.…”