Editing Planar graph (section)

=== Coin graphs ===
{{main|Circle packing theorem}}
[[File:Circle packing theorem K5 minus edge example.svg|thumb|Example of the circle packing theorem on {{math|''K''{{sup| −}}{{sub|5}}}}, the complete graph on five vertices, minus one edge.]]

We say that two circles drawn in a plane ''kiss'' (or ''[[Osculating circle|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 segment]]s 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.