Crossings between non-homotopic edges

TL;DR

This paper investigates the crossing number bounds in non-homotopic multigraphs, establishing tight asymptotic lower bounds for the number of crossings based on the number of vertices and edges.

Contribution

It introduces the concept of non-homotopic multigraphs and proves tight bounds on their crossing numbers, extending understanding of graph drawing complexities.

Findings

Crossing number exceeds c(m^2)/n for non-homotopic multigraphs with m>4n edges.

Bound is tight up to a polylogarithmic factor.

Lower bound is not asymptotically sharp as n is fixed and m grows.

Abstract

We call a multigraph {\em non-homotopic} if it can be drawn in the plane in such a way that no two edges connecting the same pair of vertices can be continuously transformed into each other without passing through a vertex, and no loop can be shrunk to its end-vertex in the same way. It is easy to see that a non-homotopic multigraph on $n>1$ vertices can have arbitrarily many edges. We prove that the number of crossings between the edges of a non-homotopic multigraph with $n$ vertices and $m>4n$ edges is larger than $c\frac{m^2}{n}$ for some constant $c>0$, and that this bound is tight up to a polylogarithmic factor. We also show that the lower bound is not asymptotically sharp as $n$ is fixed and $m$ tends to infinity.