  Every graph $G$ can be represented by a collection of equi-radii spheres in a
$d$-dimensional metric $\Delta$ such that there is an edge $uv$ in $G$ if and
only if the spheres corresponding to $u$ and $v$ intersect. The smallest
integer $d$ such that $G$ can be represented by a collection of spheres (all of
the same radius) in $\Delta$ is called the sphericity of $G$, and if the
collection of spheres are non-overlapping, then the value $d$ is called the
contact-dimension of $G$. In this paper, we study the sphericity and contact
dimension of the complete bipartite graph $K_{n,n}$ in various $L^p$-metrics
and consequently connect the complexity of the monochromatic closest pair and
bichromatic closest pair problems.
