On the density of non-simple 3-planar graphs

Abstract

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 [20] 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 [19] 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.