Unit distance graphs with few crossings per edge

TL;DR

This paper studies the maximum number of edges in unit distance graphs with limited crossings per edge, providing improved bounds for cases where each edge has at most one or two crossings, advancing understanding of geometric graph density.

Contribution

It improves upper bounds for 1-planar unit distance graphs and establishes the first non-trivial bounds for 2-planar cases, also providing new lower bound constructions.

Findings

For k=1, u_1(n) ≤ 3n - c√n, tight up to constant c.

For k=2, u_2(n) ≤ 4n - 8.

Constructed lower bounds for u_2(n) showing it exceeds u_0(n) by c√n.

Abstract

A graph is called a $k$-planar unit distance graph if it can be drawn in the plane such that every edge is a unit line segment and is involved in at most $k$ crossings. We investigate $u_k(n)$, the maximum number of edges of such graphs on $n$ vertices. For $k=1$, we improve the best known upper bound, by showing that $u_1(n) \leq 3n - c\sqrt{n}$ for some constant $c>0$. This bound is tight up to the value of the constant $c$. For $k=2$, we establish the first non-trivial upper bound by proving that $u_2(n) \leq 4n - 8$. Regarding lower bounds we give a construction for $k=2$ that shows $u_2(n) \geq u_0(n) + c\sqrt{n}$ if $n$ is sufficiently large.