On the crossing profile of rectilinear drawings of $K_n$

TL;DR

This paper introduces the crossing profile of graph drawings, focusing on rectilinear drawings of $K_n$, and characterizes the asymptotic behavior of the maximum and minimum sums of the profile's entries.

Contribution

It is the first to analyze the entire crossing profile sequence, providing bounds and characterizations for rectilinear drawings of complete graphs.

Findings

Existence of drawings with specific crossing profile entries of magnitude Ω(n).

Possibility of having zero entries in the crossing profile for large n.

Asymptotic characterization of sums of crossing profile entries.

Abstract

We introduce the \textit{crossing profile} of a drawing of a graph. This is a sequence of integers whose $(k+1)^{\text{th}}$ entry counts the number of edges in the drawing which are involved in exactly $k$ crossings. The first and second entries of this sequence (which count uncrossed edges and edges with one crossing, respectively) have been studied by multiple authors. However, to the best of our knowledge, we are the first to consider the entire sequence. Most of our results concern crossing profiles of rectilinear drawings of the complete graph $K_n$. We show that for any $k\leq (n-2)^2/4$ there is such a drawing for which the $k^{\text{th}}$ entry of the crossing profile is of magnitude $\Omega(n)$. On the other hand, we prove that for any $k \geq 1$ and any sufficiently large $n$, the $k^{\text{th}}$ entry can also be made to be $0$. As our main result, we essentially characterize the asymptotic behavior of both the maximum and minimum values that the sum of the first $k$ entries of the crossing profile might achieve. Our proofs are elementary and rely mostly on geometric constructions and classical results from discrete geometry and geometric graph theory.