TITLE: Extensions of the Koebe–Andreev–Thurston theorem to sphere packing?
QUESTION [12 upvotes]: 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!

REPLY [13 votes]: I don't know of any results on this. There are some reformulations in terms of Gram matrices, but I don't know if they help.
For $K_6$, there is no realization by touching spheres. Suppose you had such a realization. The property of having tangent spheres realizing a graph is invariant under Mobius transformations. In particular, one may perform a Mobius transformation sending the tangency between two spheres to infinity in $\mathbb{R}^3$. The two spheres are sent to parallel planes, and the other 4 spheres are tangent to both of these planes and to each other. In particular, the midplane between these two planes intersects the other 4 spheres in 4 tangent circles of equal radius. But 4 equal radius circles in the plane cannot be simultaneously tangent. So $K_6$ cannot be realized by tangent spheres.

REPLY [8 votes]: According to corollary 4.6 of "Representing Graphs by Disks and Balls (a survey of recognition-complexity results)" these graphs are NP hard to recognize.

REPLY [7 votes]: 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.