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The circle packing theorem describes the possible patterns of tangent circles among non-overlapping circles in the plane. A circle packing is a collection of circles whose union is connected and whose interiors are disjoint. The intersection graph of a circle packing, called a coin graph, is the graph having a vertex for each circle, and an edge for every pair of circles that are tangent. Coin graphs are always connected, simple, and planar. The circle packing theorem states that these are the only requirements for a graph to be a coin graph: every finite connected simple planar graph has a circle packing in the plane whose intersection graph is isomorphic to .
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