Unit distance graphs with few crossings per edge

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.