Impossible patterns of sphere tangencies

My latest preprint, “Tangent spheres and integer distances” (arXiv:2606.18569, to appear at CCCG), involves the patterns of external tangencies of circles, spheres or higher-dimensional hyperspheres. You can make a graph whose vertices are a given set of spheres and whose edges are pairs of externally-tangent spheres, and I’d like to understand which graphs are possible. By the circle packing theorem, any planar graph can be represented by interior-disjoint circles in this way, but here I’m not requiring disjointness. So, for instance, you can represent any unit distance graph in the plane (such as the Petersen graph below) by expanding each vertex of the graph into a unit-diameter circle. In three dimensions, you can produce spheres whose tangencies form arbitrarily large complete bipartite graphs \(K_{n,n}\) by placing \(n\) spheres interior to a torus, like a ball-bearing race or those cat toys with balls rolling around a toroidal track, and another \(n\) spheres tangent to them,…