Separation dimension of bounded degree graphs

TL;DR

This paper investigates the separation dimension of bounded degree graphs, establishing an upper bound related to the maximum degree and showing that most such graphs have a high separation dimension, close to the upper bound.

Contribution

It provides a new upper bound on the separation dimension for graphs with bounded degree and demonstrates the bound's near tightness for most regular graphs.

Findings

Separation dimension of bounded degree graphs is at most exponential in the iterated logarithm of the degree.

Almost all d-regular graphs have a separation dimension at least half of d.

The established bound is nearly tight for large degrees.

Abstract

The 'separation dimension' of a graph $G$ is the smallest natural number $k$ for which the vertices of $G$ can be embedded in $\mathbb{R}^k$ such that any pair of disjoint edges in $G$ can be separated by a hyperplane normal to one of the axes. Equivalently, it is the smallest possible cardinality of a family $\mathcal{F}$ of total orders of the vertices of $G$ such that for any two disjoint edges of $G$, there exists at least one total order in $\mathcal{F}$ in which all the vertices in one edge precede those in the other. In general, the maximum separation dimension of a graph on $n$ vertices is $\Theta(\log n)$. In this article, we focus on bounded degree graphs and show that the separation dimension of a graph with maximum degree $d$ is at most $2^{9log^{\star} d} d$. We also demonstrate that the above bound is nearly tight by showing that, for every $d$, almost all $d$-regular graphs have separation dimension at least $\lceil d/2\rceil$.