A first view on the density of 5-planar graphs

TL;DR

This paper investigates the maximum edge density of 5-planar graphs, providing new bounds and structural insights, and introduces a simplified discharging technique applicable to related graph classes.

Contribution

It introduces a simplified discharging method to analyze 5-planar graphs, establishing tighter density bounds and revealing structural differences from smaller k-planar graphs.

Findings

Graphs with simple 5-planar drawings have at most 7(n-2) edges.

The new bound improves the previous estimate of approximately 8.3n.

The technique also applies to 4- and 6-planar graphs, showing its versatility.

Abstract

A key concept for many graph layout algorithms is planarity, a graph property that allows to draw vertices and edges crossing-free in the plane. Important is the generalization to $k$-planar graphs, which can be drawn in the plane with at most $k > 0$ crossings per edge. One of the basic graph properties that have been explored for those graph classes is the maximum edge density, i.e., the maximum number of edges a $k$-planar graph on $n$ vertices may have. While there are numerous results for the classes of $1$- and $2$-planar graphs, there are few results for increasing $k=3$ or $4$ due to the complex graph structures. We make a first step towards even larger $k>4$ exploring the class of $5$-planar graphs. While our main tool is still a discharging technique, a better understanding of the structure of the denser parts leads to corresponding density bounds in a much simpler way. We first apply a simplified version of our technique to outer $5$-planar graphs and surprisingly observe that the structure of maximally dense (general) $5$-planar graphs differs from the known uniform structure of maximally dense $k$-planar graphs for smaller $1 \leq k \leq 4$. As the central result of this paper, we then show that graphs that admit a simple 5-planar drawing have at most $7(n-2)$ edges, drastically improving the previous best bound of $\approx8.3n$. This even implies a small improvement of the leading constant in the Crossing Lemma $cr(G) \ge c \frac{m^3}{n^2}$ from $c=\frac{1}{27.48}$ to $c=\frac{1}{27.3}$. To demonstrate the potential of our new technique, we also apply it to 4-planar and 6-planar graphs.