Improving the Crossing Lemma by Characterizing Dense 2-Planar and 3-Planar Graphs

TL;DR

This paper improves the Crossing Lemma bounds by analyzing dense 2-planar and 3-planar graphs, introducing new local configuration restrictions that lead to tighter crossing number lower bounds.

Contribution

It introduces novel bounds for the crossing number by characterizing forbidden local configurations in dense 2- and 3-planar graphs, enhancing classical crossing number estimates.

Findings

Derived new lower bounds for crossing numbers in specific graph density ranges.

Identified fewer edges in 2- and 3-planar graphs under certain local configuration restrictions.

Improved the constant in the Crossing Lemma from 1/29 to over 1/27.48.

Abstract

The classical Crossing Lemma by Ajtai et al.~and Leighton from 1982 gave an important lower bound of $c \frac{m^3}{n^2}$ for the number of crossings in any drawing of a given graph of $n$ vertices and $m$ edges. The original value was $c= 1/100$, which then has gradually been improved. Here, the bounds for the density of $k$-planar graphs played a central role. Our new insight is that for $k=2,3$ the $k$-planar graphs have substantially fewer edges if specific local configurations that occur in drawings of $k$-planar graphs of maximum density are forbidden. Therefore, we are able to derive better bounds for the crossing number $\text{cr}(G)$ of a given graph $G$. In particular, we achieve a bound of $\text{cr}(G) \ge \frac{37}{9}m-\frac{155}{9}(n-2)$ for the range of $5n < m \le 6n$, while our second bound $\text{cr}(G) \ge 5m - \frac{203}{9}(n-2)$ is even stronger for larger $m>6n$. For $m > 6.77n$, we finally apply the standard probabilistic proof from the BOOK and obtain an improved constant of $c>1/27.48$ in the Crossing Lemma. Note that the previous constant was $1/29$. Although this improvement is not too impressive, we consider our technique as an important new tool, which might be helpful in various other applications.