Boxicity and Cubicity of Product Graphs

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.