Email this article Complete bipartite graphs flexible in the plane
- Authors: Kovalev M.D.1, Orevkov S.Y.2,3
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Affiliations:
- Lomonosov Moscow State University, Faculty of Mechanics and Mathematics
- Steklov Mathematical Institute of Russian Academy of Sciences
- Institut de Mathématiques de Toulouse
- Issue: Vol 214, No 10 (2023)
- Pages: 44-70
- Section: Articles
- URL: https://journal-vniispk.ru/0368-8666/article/view/140516
- DOI: https://doi.org/10.4213/sm9867
- ID: 140516
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Abstract
A complete bipartite graph $K_{3,3}$, considered as a planar linkage with joints at the vertices and with rods as edges, is in general inflexible, that is, it admits only motions as a whole. Two types of its paradoxical mobility were found by Dixon in 1899. Later on, in a series of papers by several different authors the question of the flexibility of $K_{m,n}$ was solved for almost all pairs $(m,n)$. We solve it for all complete bipartite graphs in the Euclidean plane, as well as on the sphere and hyperbolic plane. We give independent self-contained proofs without extensive computations, which are almost the same in the Euclidean, hyperbolic and spherical cases.
About the authors
Mikhail Dmitrievich Kovalev
Lomonosov Moscow State University, Faculty of Mechanics and Mathematics
Author for correspondence.
Email: mdkovalev@mtu-net.ru
Doctor of physico-mathematical sciences, Professor
Stepan Yur'evich Orevkov
Steklov Mathematical Institute of Russian Academy of Sciences; Institut de Mathématiques de Toulouse
Email: orevkov@math.ups-tlse.fr
Candidate of physico-mathematical sciences, Senior Researcher
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