Crossing Number of 3-Plane Drawings
Miriam Goetze, Michael Hoffmann, Ignaz Rutter, Torsten Ueckerdt
TL;DR
This work studies $3$-plane drawings, where each edge has at most three crossings, and proves a tight bound $|X| \le 5.5\,(|V|-2)$ on the number of crossings in such drawings, thereby obtaining $\mathrm{cr}(G) \le 5.5\,(n-2)$ for $3$-planar graphs and an improved edge bound. The authors deploy the Density Formula for topological graph drawings with $t=5$ and construct a linear program that ties together vertex, edge, crossing, and cell counts; solving it yields the stated bound and, analogously, a bound on $|E|$. A key innovation is the introduction of configurations and trails to analyze the interplay of cells, enabling new inequalities that feed the Density Formula. The approach tightens known results (from $6.6n$) and, via corollaries, yields improved bounds for restricted classes such as $C_3$-free or girth-5 $3$-planar graphs, contributing a versatile framework for crossing-number analysis in topological graph drawings.
Abstract
We study 3-plane drawings, that is, drawings of graphs in which every edge has at most three crossings. We show how the recently developed Density Formula for topological drawings of graphs (KKKRSU GD 2024) can be used to count the crossings in terms of the number $n$ of vertices. As a main result, we show that every 3-plane drawing has at most $5.5(n-2)$ crossings, which is tight. In particular, it follows that every 3-planar graph on $n$ vertices has crossing number at most $5.5n$, which improves upon a recent bound (BBBDHKMOW GD 2024) of $6.6n$. To apply the Density Formula, we carefully analyze the interplay between certain configurations of cells in a 3-plane drawing. As a by-product, we also obtain an alternative proof for the known statement that every 3-planar graph has at most $5.5(n-2)$ edges.