On the crossing profile of rectilinear drawings of $K_n$

TL;DR

This work introduces the crossing profile cp(D) of a drawing, a detailed edge-crossing distribution capturing how many edges participate in exactly $k$ crossings. Focusing on rectilinear drawings of the complete graph $K_n$, it delivers both lower and upper bounds on the numbers of edges with at most $k$ crossings, and on the number of edges with exactly $k$ crossings, via elementary geometric constructions and the cutting lemma. The authors prove a near-complete asymptotic picture: (i) for $k\le (n-2)^2/4$, there exist drawings with $e_k(K_n)$ linear in $n$; (ii) for every fixed $k\ge1$ and large $n$, $e_k(K_n)$ can be forced to $0$; and (iii) the sums $S_k(K_n)$ have tight asymptotics for both their maximum and minimum across all rectilinear drawings, with precise regimes depending on how $k$ scales with $n$. These results illuminate the fine-grained distribution of crossings and connect to broader themes in geometric graph theory, such as $k$-planarity and $k$-set phenomena, using accessible geometric tools like the cutting lemma. The findings have implications for understanding the structure of dense geometric graphs and for illustrating how crossing distributions constrain edge layouts in rectilinear drawings.

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 $Ω(n)$. 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.

Figures (15)

• Figure 1: The positions of the vertices in a rectilinear drawing of $K_7$ with $12$ edges that are involved in no crossings.
• Figure 2: A configuration of $12$ points for $m=6$ and $k = 19$.
• Figure 3: A configuration of $12$ points for $m=6$ and $k = 23$.
• Figure 4: Grouping together $5$ copies of the $2m$-point configuration described above. Some leftover nodes have been placed near the center of the circle.
• Figure 5: Construction for $n = 16$ where edges with exactly one crossing have been marked in red. Note, for example, that $\overline{AQ_6}$ is the only edge crossing $\overline{P_2Q_2}$.

Theorems & Definitions (39)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Theorem 2.5: Crossing lemma, Ajtai et al. and Leighton, crossinglemmacrossinglemma2
• Definition 2.6
• Definition 2.7
• Definition 2.8
• Theorem 2.9: Ringel, ringel1964extremal
• Theorem 2.10: Harborth and Mengersen, harborth1974edges