On the Density of 3-Planar Graphs

A k-planar graph is a graph that can be drawn in the plane such that every edge is crossed at most k times. For k ≤ 4, Pach and Tóth [19] proved a bound of (k + 3)(n− 2) on the total number of edges of a k-planar graph, which is tight for k = 1, 2. For k = 3, the bound of 6n− 12 has been improved to 11 2 n− 11 in [18] and has been shown to be optimal up to an additive constant for simple graphs. In this paper, we prove that the bound of 11 2 n−11 edges also holds for non-simple 3-planar graphs that admit drawings in which non-homotopic parallel edges and self-loops are allowed. Based on this result, a characterization of optimal 3-planar graphs (that is, 3-planar graphs with n vertices and exactly 11 2 n − 11 edges) might be possible, as to the best of our knowledge the densest known simple 3-planar is not known to be optimal.