Improvements on the density of maximal 1-planar graphs

TL;DR

This paper improves the known lower bounds on the number of edges in maximal 1-planar and 1-plane graphs, narrowing the gap between the minimal and maximal edge counts.

Contribution

It provides new, tighter lower bounds of n/9 7 2.22n for the edge counts in maximal 1-planar and 1-plane graphs, advancing understanding of their density.

Findings

Lower bounds on edges increased to 20n/9 7 2.22n

Previous bounds were 28/13 n 7 2.15n and 2.1n

Results improve theoretical understanding of graph density limits

Abstract

A graph is 1-planar if it can be drawn in the plane such that each edge is crossed at most once. A graph, together with a 1-planar drawing is called 1-plane. Brandenburg et al. showed that there are maximal 1-planar graphs with only $\frac{45}{17}n + O(1)\approx 2.647n$ edges and maximal 1-plane graphs with only $\frac{7}{3}n+O(1)\approx 2.33n$ edges. On the other hand, they showed that a maximal 1-planar graph has at least $\frac{28}{13}n-O(1)\approx 2.15n-O(1)$ edges, and a maximal 1-plane graph has at least $2.1n-O(1)$ edges. We improve both lower bounds to $\frac{20n}{9}\approx 2.22n$.