On the Maximum Number of Edges in 1-Planar Unit Distance Graphs

A matchstick graph is a plane graph whose edges are represented by line segments of unit length. For this class, Harborth conjectured an upper bound on the maximum number of edges, which was later confirmed by Lavollée and Swanepoel. In this talk we consider 1-planar unit distance graphs: graphs that admit a drawing in the plane where all edges are unit-length line segments and each edge is involved in at most one crossing. Previously, the best constructions for achieving the maximum number of edges on n vertices were matchstick graphs. We introduce three new constructions of 1-planar unit distance graphs that yield more edges than matchstick graphs on the same number of vertices. This result answers an open question posed by Gehér and Tóth.