Description
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 floor of 3n - \sqrt{12n - 3} edges. Recently his conjecture was settled by Lavollée and Swanepoel. In this talk 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}/10 edges. Joint work with Géza Tóth.