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].")
<br />&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
<img src="https://www.ics.uci.edu/~eppstein/0xDE/KesPacPal-GD-10.png" width="250" alt="Circle Packing" />
<br />
<sub>(Image due to David Eppstein, [here][2].)</sub>



> 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&mdash;Thanks for pointers!


  [1]: http://en.wikipedia.org/wiki/Circle_packing_theorem
  [2]: http://11011110.livejournal.com/205447.html