Have the graphs representable by touching tetrahedra been explored?

Let $\cal T$ be a collection of tetrahedra in $\mathbb{R}^3$
with pairwise disjoint interiors.
Define a graph $G_{\cal T}$ to have a node for each tetrahedron
in $\cal T$, and an arc between $T_1$ and $T_2$ if those two
tetrahedra share one or more boundary points:
$T_1 \cap T_2 \neq \varnothing$.

[*Added*]
I neglected to add the significant qualification (of most
interest to me) that each arc of $G_{\cal T}$
should be able to be associated with a unique point. My oversight only became clear
with Aaron's example.  Apologies!  Of course one can make many definitions (such as Igor's variation), and all are of interest.

> Which graphs $G$ are equal to $G_{\cal T}$ for some $\cal T$?

For example,
$K_6$ is a touching-tetrahedra graph:
<br />
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
<img src="http://cs.smith.edu/~orourke/MathOverflow/TouchingTetrahedra.jpg" alt="TouchingTetrahedra" />
<br />
In contrast, responses to an earlier MO question,
"<a href="https://mathoverflow.net/questions/85547/">Extensions of the Koebe–Andreev–Thurston theorem to sphere packing?</a>"
showed that $K_6$ is not a ball-touching graph.

I know triangle-touching graphs in $\mathbb{R}^2$ have been
studied, often called *triangle contact representations*,
e.g., the recent paper by
Gonçalves, Lévêque, and Pinlou,
"<a href="http://hal-lirmm.ccsd.cnrs.fr/lirmm-00620728/fr">Triangle Contact Representations and Duality</a>."
But I haven't found literature on the generalization to tetrahedra.

I would be interested in any pointers to the literature,
or classes of graphs that either are or are not
touching-tetrahedra graphs.  E.g., is $K_7$ a touching-tetrahedra graph?
Thanks!