1-planar unit distance graphs

TL;DR

This paper investigates 1-planar unit distance graphs, establishing an upper bound on the number of edges relative to vertices, and explores generalizations like k-planar and k-quasiplanar graphs.

Contribution

It introduces bounds on edges for 1-planar unit distance graphs and extends the analysis to k-planar and k-quasiplanar cases, advancing understanding of geometric graph constraints.

Findings

Maximum edges for 1-planar unit distance graphs are at most 3n - n^(1/4)/15.

Bound is nearly tight, indicating close to optimality.

Explores generalizations to k-planar and k-quasiplanar graphs.

Abstract

A matchstick graph is a plane graph with edges drawn as unit distance line segments. This class of graphs was introduced by Harborth who conjectured that a matchstick graph on $n$ vertices can have at most $\lfloor 3n - \sqrt{12n - 3}\rfloor$ edges. Recently, his conjecture was settled by Lavoll\'ee and Swanepoel. In this paper we consider $1$-planar unit distance graphs. We say that a graph is a $1$-planar unit distance graph if it can be drawn in the plane such that all edges are drawn as unit distance line segments while each of them are involved in at most one crossing. We show that such graphs on $n$ vertices can have at most $3n-\sqrt[4]{n}/15$ edges, which is almost tight. We also investigate some generalizations, namely $k$-planar and $k$-quasiplanar unit distance graphs.