Describes the possible tangency relations between circles with disjoint interiors
"Coin graph" redirects here. For contact graphs of equal-radius disks, see penny graph.
A circle packing for a five-vertex planar graph
The circle packing theorem (also known as the Koebe–Andreev–Thurston theorem) describes the possible tangency relations between circles in the plane whose interiors are disjoint. A circle packing is a connected collection of circles (in general, on any Riemann surface) whose interiors are disjoint. The intersection graph of a circle packing is the graph having a vertex for each circle, and an edge for every pair of circles that are tangent. If the circle packing is on the plane, or, equivalently, on the sphere, then its intersection graph is called a coin graph; more generally, intersection graphs of interior-disjoint geometric objects are called tangency graphs or contact graphs. 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:
Circle packing theorem: For every connected simple planar graph G there is a circle packing in the plane whose intersection graph is (isomorphic to) G.
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The circlepackingtheorem (also known as the Koebe–Andreev–Thurston theorem) describes the possible tangency relations between circles in the plane whose...
In geometry, circlepacking is the study of the arrangement of circles (of equal or varying sizes) on a given surface such that no overlapping occurs...
interiors, by making a vertex for each circle and an edge for each pair of circles that kiss. The circlepackingtheorem, first proved by Paul Koebe in 1936...
circle packing theorem. It was written by Kenneth Stephenson and published in 2005 by the Cambridge University Press. Circlepackings, as studied in this...
is the three-dimensional equivalent of the circlepacking in a circle problem in two dimensions. Best packing of m>1 equal spheres in a sphere setting a...
set of Kleinian groups; see also Circlepackingtheorem. The circles of Apollonius may also denote three special circles C 1 , C 2 , C 3 {\displaystyle...
This concept generalizes the circlepackings described by the circlepackingtheorem, in which specified pairs of circles are tangent to each other. Although...
tangent circles whose tangencies represent a given Apollonian network forms a simple instance of the Koebe–Andreev–Thurston circle-packingtheorem, which...
appear in the ring lemma, used to prove connections between the circlepackingtheorem and conformal maps. The Fibonacci numbers are important in computational...
The circlepackingtheorem states that every planar graph can be represented as a contact graph of circles. The contact graphs of unit circles are called...
mathematician and astronomer Johannes Kepler, is a mathematical theorem about sphere packing in three-dimensional Euclidean space. It states that no arrangement...
Integral Apollonian circlepacking defined by circle curvatures of (−3, 5, 8, 8) Integral Apollonian circlepacking defined by circle curvatures of (−12...
the plane is a unit disk graph. The Circlepackingtheorem states that the intersection graphs of non-crossing circles are exactly the planar graphs. Scheinerman's...
even when the locations are fixed. Circlepacking in a rectangle Square packing in a square De Bruijn's theorem: packing congruent rectangular bricks of...
unit disks in the plane. A circle graph is the intersection graph of a set of chords of a circle. The circlepackingtheorem states that planar graphs...
lengths. One stronger form of the circlepackingtheorem, on representing planar graphs by systems of tangent circles, states that every polyhedral graph...
geometry of circlepackings in the Euclidean plane, the ring lemma gives a lower bound on the sizes of adjacent circles in a circlepacking. The lemma...
Closed Steiner chains are the systems of circles obtained as the circlepackingtheorem representation of a bipyramid. Annular Steiner chains n = 3 n =...
planar graph. Alternatively, by the circlepackingtheorem, any planar graph may be represented as a collection of circles, any two of which cross if and only...
rediscovered Descartes' theorem on the radii of mutually tangent quadruples of circles. Any triangle has three externally tangent circles centered at its vertices...
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