Twisted Ways to Find Plane Structures in Simple Drawings of Complete Graphs

TL;DR

This paper introduces generalized twisted drawings of complete graphs, proving they contain many disjoint edges and long plane paths, thereby improving known lower bounds and revealing new structural properties.

Contribution

It defines a new class of simple graph drawings called generalized twisted drawings and establishes improved lower bounds for disjoint edges and path lengths in such drawings.

Findings

Every simple drawing of a complete graph contains (n^{1/2}) disjoint edges.

Such drawings contain a plane path of length (rac{\u221d n}{\u221d })

Structural properties and characterizations of generalized twisted drawings are presented.

Abstract

Simple drawings are drawings of graphs in which the edges are Jordan arcs and each pair of edges share at most one point (a proper crossing or a common endpoint). We introduce a special kind of simple drawings that we call generalized twisted drawings. A simple drawing is generalized twisted if there is a point $O$ such that every ray emanating from $O$ crosses every edge of the drawing at most once and there is a ray emanating from $O$ which crosses every edge exactly once. Via this new class of simple drawings, we show that every simple drawing of the complete graph with $n$ vertices contains $\Omega(n^{\frac{1}{2}})$ pairwise disjoint edges and a plane path of length $\Omega(\frac{\log n }{\log \log n})$. Both results improve over previously known best lower bounds. On the way we show several structural results about and properties of generalized twisted drawings. We further present different characterizations of generalized twisted drawings, which might be of independent interest.