Planar projections of graphs

TL;DR

This paper introduces a novel graph representation using multi-dimensional embeddings and axis-parallel projections, providing bounds and optimality results for complete graphs and graphs with specific properties.

Contribution

It presents new bounds for representing graphs via projections onto multiple planes, including optimal bounds for complete graphs and graphs with maximum degree 5.

Findings

Complete graph on n vertices has a representation in approximately sqrt(n/2) planes.

Graphs with 6n-15 edges can be projected onto two planes in 3D, which is optimal.

Graphs with maximum degree 5 have a 3D plane-projectable representation.

Abstract

We introduce and study a new graph representation where vertices are embedded in three or more dimensions, and in which the edges are drawn on the projections onto the axis-parallel planes. We show that the complete graph on $n$ vertices has a representation in $\lceil \sqrt{n/2}+1 \rceil$ planes. In 3 dimensions, we show that there exist graphs with $6n-15$ edges that can be projected onto two orthogonal planes, and that this is best possible. Finally, we obtain bounds in terms of parameters such as geometric thickness and linear arboricity. Using such a bound, we show that every graph of maximum degree 5 has a plane-projectable representation in 3 dimensions.