In tetrahedron contact graphs, the nodes are represented by interior
disjoint tetrahedra and the arcs are represented by contacts between
pairs of tetrahedra such that no three tetrahedra touch at a common
point. We prove that planar graphs, graphs with at most 7 nodes,
complete graphs with at most 10 nodes, complete bipartite and complete
tripartite graphs can be realized with general tetrahedra. For regular
tetrahedra with unit edge length, not even all binary trees can be
realized; but graphs with 5 nodes and all complete bipartite graphs
K_{m,n} with m,n at most 3 can be realized. Finally, for both general
and unit-regular tetrahedra we study a restricted variant where two
tetrahedra can touch only through a common vertex.