Convex Hull Thrackles

TL;DR

This paper investigates convex hull thrackles, a generalization of linear thrackles, proving bounds on the number of convex hulls relative to vertices in convex position and generally.

Contribution

It establishes upper bounds on the number of convex hulls in convex hull thrackles, including a tight bound for convex position and a general bound of twice the vertices.

Findings

Number of convex hulls ≤ number of vertices in convex position.

Existence of constructions with one more convex hull than vertices.

Number of convex hulls ≤ twice the number of vertices in general.

Abstract

A \emph{thrackle} is a graph drawn in the plane so that every pair of its edges meet exactly once, either at a common end vertex or in a proper crossing. Conway's thrackle conjecture states that the number of edges is at most the number of vertices. It is known that this conjecture holds for linear thrackles, i.e., when the edges are drawn as straight line segments. We consider \emph{convex hull thrackles}, a recent generalization of linear thrackles from segments to convex hulls of subsets of points. We prove that if the points are in convex position then the number of convex hulls is at most the number of vertices, but in general there is a construction with one more convex hull. On the other hand, we prove that the number of convex hulls is always at most twice the number of vertices.