Planar graph
Coin graphs

We say that two circles drawn in a plane kiss (or osculate) whenever they intersect in exactly one point. A "coin graph" is a graph formed by a set of circles, no two of which have overlapping interiors, by making a vertex for each circle and an edge for each pair of circles that kiss. The circle packing theorem, first proved by Paul Koebe in 1936, states that a graph is planar if and only if it is a coin graph.

This result provides an easy proof of Fáry's theorem, that every simple planar graph can be embedded in the plane in such a way that its edges are straight line segments that do not cross each other. If one places each vertex of the graph at the center of the corresponding circle in a coin graph representation, then the line segments between centers of kissing circles do not cross any of the other edges.
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