Boxicity and Cubicity of Product Graphs

TL;DR

This paper investigates the boxicity and cubicity of product graphs, providing bounds and growth rates for powers of graphs under various graph products, revealing surprising and complex behaviors.

Contribution

It offers new estimates and growth bounds for the boxicity and cubicity of Cartesian, strong, and direct product graphs, especially for graph powers.

Findings

Boxicity of Cartesian powers grows as O(log d / log log d)

Cubicity of Cartesian powers grows as θ(d / log d)

No sublinear bounds exist for boxicity growth in strong and direct powers

Abstract

The 'boxicity' ('cubicity') of a graph G is the minimum natural number k such that G can be represented as an intersection graph of axis-parallel rectangular boxes (axis-parallel unit cubes) in $R^k$. In this article, we give estimates on the boxicity and the cubicity of Cartesian, strong and direct products of graphs in terms of invariants of the component graphs. In particular, we study the growth, as a function of $d$, of the boxicity and the cubicity of the $d$-th power of a graph with respect to the three products. Among others, we show a surprising result that the boxicity and the cubicity of the $d$-th Cartesian power of any given finite graph is in $O(\log d / \log\log d)$ and $\theta(d / \log d)$, respectively. On the other hand, we show that there cannot exist any sublinear bound on the growth of the boxicity of powers of a general graph with respect to strong and direct products.