Annular and pants thrackles

TL;DR

This paper proves Conway's Thrackle Conjecture for a special class of thrackle drawings where vertices lie on at most three boundary domains, and provides detailed descriptions for annular and pants thrackles.

Contribution

It establishes the conjecture for thrackles with vertices on up to three boundary domains and characterizes annular and pants thrackle drawings.

Findings

Proved the conjecture for thrackles with vertices on up to three boundary domains.

Provided detailed descriptions of annular and pants thrackle drawings.

Characterized the structure of thrackles in these specific cases.

Abstract

A thrackle is a drawing of a graph in which each pair of edges meets precisely once. Conway's Thrackle Conjecture asserts that a thrackle drawing of a graph on the plane cannot have more edges than vertices. We prove the Conjecture for thrackle drawings all of whose vertices lie on the boundaries of $d \le 3$ connected domains in the complement of the drawing. We also give a detailed description of thrackle drawings corresponding to the cases when $d=2$ (annular thrackles) and $d=3$ (pants thrackles).