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.

Figures (7)

• Figure 1: All types of cells with size at most $5$ that can occur in the planarization of a straight line drawing of a graph.
• Figure 2: The unique configuration of two cells of size $5$ with a common crossing.
• Figure 4: The four types of bad triangles. Edges in $E_1$ are drawn with dashed lines.
• Figure 5: $\Phi$ is a bad triangle with halfedges $\alpha$ and $\beta$. Let $\Psi$ be the helper of $\Phi$. Then (a) $\Psi$ cannot be a triangle, and (b) $\Psi$ cannot be a quadrilateral.
• Figure 6: The basis of the construction. It has $9$ vertices and $18$ edges.

Theorems & Definitions (26)

• Theorem 1
• Theorem 2
• Theorem 3
• Remark 1
• proof : Proof of \ref{['thm_1planar']}
• Theorem 4: Density formula KKKRSU23
• Claim 1
• proof
• Claim 2
• proof