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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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The non-solvability by radicals of generic 3-connected planar Laman graphs
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by J. C. Owen and S. C. Power PDF
Trans. Amer. Math. Soc. 359 (2007), 2269-2303 Request permission

Abstract:

We show that planar embeddable $3$-connected Laman graphs are generically non-soluble. A Laman graph represents a configuration of points on the Euclidean plane with just enough distance specifications between them to ensure rigidity. Formally, a Laman graph is a maximally independent graph, that is, one that satisfies the vertex-edge count $2v - 3 = e$ together with a corresponding inequality for each subgraph. The following main theorem of the paper resolves a conjecture of Owen (1991) in the planar case. Let $G$ be a maximally independent $3$-connected planar graph, with more than 3 vertices, together with a realisable assignment of generic distances for the edges which includes a normalised unit length (base) edge. Then, for any solution configuration for these distances on a plane, with the base edge vertices placed at rational points, not all coordinates of the vertices lie in a radical extension of the distance field.
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Additional Information
  • J. C. Owen
  • Affiliation: D-Cubed Ltd., Park House, Cambridge CB3 0DU, United Kingdom
  • Email: john.owen@d-cubed.co.uk
  • S. C. Power
  • Affiliation: Department of Mathematics and Statistics, Lancaster University, Lancaster, LA1 4YF, United Kingdom
  • MR Author ID: 141635
  • Email: s.power@lancaster.ac.uk
  • Received by editor(s): October 16, 2003
  • Received by editor(s) in revised form: March 15, 2005
  • Published electronically: October 16, 2006
  • © Copyright 2006 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 359 (2007), 2269-2303
  • MSC (2000): Primary 68U07, 12F10, 05C40; Secondary 52C25
  • DOI: https://doi.org/10.1090/S0002-9947-06-04049-9
  • MathSciNet review: 2276620