[Math] Extensions of the Koebe–Andreev–Thurston theorem to sphere packing

Question

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.")

         

_{(source: uci.edu)}

_{(Image due to David Eppstein, here.)}

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!

Answer 1

Yes, certain restrictions are well known. One reference is Kuperberg & Schramm here ("Average kissing numbers for non-congruent sphere packings", 1994) which says that such graphs would have to have average degree <15. A more recent reference is Benjamini & Schramm here ("Lack of Sphere Packing of Graphs via Non-Linear Potential Theory", 2009) which shows that certain low degree infinite graphs are not realizable this way.