The Koebe–Andreev–Thurston theorem states that any planar graph can be represented 
"in such a way that its vertices correspond to disjoint disks, which touch if and only if
the corresponding vertices are adjacent" (to quote Günter Ziegler, *Lectures on Polytopes*, Springer, 1995 p.117.
(See also the Wikipedia article, "[Circle packing theorem][1].")

> What is the corresponding statement for spheres in $\mathbb{R}^3$?
Every graph $G$ satisfying property $X$(?) can be represented by touching spheres.

This is surely known—Thanks for pointers!





  [1]: http://en.wikipedia.org/wiki/Circle_packing_theorem