On Extremal Coin Graphs, Flowers, and Their Rational Representations
Date
2009-07-06T14:43:15Z
Authors
Dunham, Jill Bigley
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Abstract
We study extremal coin graphs in the Euclidean plane on n vertices with the maximum number of edges. This is related to the unit coin graph problem first posed by Erdos in 1946, and considers coin graphs that satisfy certain conditions relating to the ratios of the possible radii of the coins in the graph. A motivating problem is a special case of a coin graph with multiple radii. We explore the algebraic equations describing a flower, the coin graph presentation of a wheel graph, and present a class of irreducible symmetric polynomials that describe the relation of the radii of each flower. These polynomials are then used to fully characterize flowers on four coins, also known as Soddy circles, with rational radii. This yields a free parametrization of all flowers on four coins with rational radii. A similar method is used to characterize all flowers on five coins with rational radii and to describe a large class of solutions for flowers on n coins.
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Coin graph, Galois theory, Diophantine equations, Plane graphs, Symmetric polynomials, Discrete geometry