Circle Packing Theorem - Generalizations of The Circle Packing Theorem

Generalizations of The Circle Packing Theorem

The circle packing theorem generalizes to graphs that are not planar. If G is a graph that can be embedded on a surface S, then there is a constant curvature Riemannian metric d on S and a circle packing on (S, d) whose contacts graph is isomorphic to G. If S is closed (compact and without boundary) and G is a triangulation of S, then (S, d) and the packing are unique up to conformal equivalence. If S is the sphere, then this equivalence is up to Möbius transformations; if it is a torus, then the equivalence is up to scaling by a constant and isometries, while if S has genus at least 2, then the equivalence is up to isometries.

Another generalization of the circle packing theorem involves replacing the condition of tangency with a specified intersection angle between circles corresponding to neighboring vertices. A particularly elegant version is as follows. Suppose that G is a finite 3-connected planar graph (that is, a polyhedral graph), then there is a pair of circle packings, one whose intersection graph is isomorphic to G, another whose intersection graph is isomorphic to the planar dual of G, and for every vertex in G and face adjacent to it, the circle in the first packing corresponding to the vertex intersects orthogonally with the circle in the second packing corresponding to the face.

Yet another variety of generalizations allow shapes that are not circles. Suppose that G = (V, E) is a finite planar graph, and to each vertex v of G corresponds a shape, which is homeomorphic to the closed unit disk and whose boundary is smooth. Then there is a packing in the plane such that if and only if and for each the set is obtained from by translating and scaling. (Note that in the original circle packing theorem, there are three real parameters per vertex, two of which describe the center of the corresponding circle and one of which describe the radius, and there is one equation per edge. This also holds in this generalization.) One proof of this generalization can be obtained by applying Koebe's original proof and the theorem of Brandt and Harrington stating that any finitely connected domain is conformally equivalent to a planar domain whose boundary components have specified shapes, up to translations and scaling.