On the Density of non-Simple 3-Planar Graphs

TL;DR

This paper extends the maximum edge bound of 3-planar graphs from simple to non-simple graphs, showing that the same density limit applies even when non-homotopic parallel edges and self-loops are allowed.

Contribution

It proves that the edge bound for simple 3-planar graphs also holds for non-simple 3-planar graphs, broadening understanding of their density limits.

Findings

The maximum number of edges in non-simple 3-planar graphs is at most (11/2)n - 11.

The bound of (11/2)n - 11 edges is tight for non-simple 3-planar graphs.

This result suggests potential for characterizing optimal 3-planar graphs beyond simple cases.

Abstract

A \emph{$k$-planar graph} is a graph that can be drawn in the plane such that every edge is crossed at most $k$ times. For $k \leq 4$, Pach and T\'oth 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 $\frac{11}{2}n-11$ and has been shown to be optimal up to an additive constant for simple graphs. In this paper, we prove that the bound of $\frac{11}{2}n-11$ 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 \emph{optimal $3$-planar graphs} (that is, $3$-planar graphs with $n$ vertices and exactly $\frac{11}{2}n-11$ edges) might be possible, as to the best of our knowledge the densest known simple $3$-planar is not known to be optimal.