Impossible patterns of sphere tangencies

🫧 Open on Bubbles 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 he