Complete bipartite graphs flexible in the plane
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2022
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| _version_ | 1866916504733220864 |
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| author | Kovalev, M. D. Orevkov, S. Yu. |
| author_facet | Kovalev, M. D. Orevkov, S. Yu. |
| contents | A complete bipartite graph $K_{3,3}$, considered as a planar linkage with joints at the vertices and with rods as edges, in general admits only motions as a whole, i.e., is inflexible. Two types of its paradoxical mobility were found by Dixon in 1899. Later on, in a series of papers by different authors, the question of flexibility of $K_{m,n}$ was solved for almost all pairs $(m,n)$. In the present paper, we solve it for all complete bipartite graphs in the Euclidean plane as well as in the sphere and in the hyperbolic plane. We give independent self-contained proofs without extensive computations which are almost the same in the Euclidean, hyperbolic and spherical cases. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2212_11231 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Complete bipartite graphs flexible in the plane Kovalev, M. D. Orevkov, S. Yu. Algebraic Geometry Metric Geometry A complete bipartite graph $K_{3,3}$, considered as a planar linkage with joints at the vertices and with rods as edges, in general admits only motions as a whole, i.e., is inflexible. Two types of its paradoxical mobility were found by Dixon in 1899. Later on, in a series of papers by different authors, the question of flexibility of $K_{m,n}$ was solved for almost all pairs $(m,n)$. In the present paper, we solve it for all complete bipartite graphs in the Euclidean plane as well as in the sphere and in the hyperbolic plane. We give independent self-contained proofs without extensive computations which are almost the same in the Euclidean, hyperbolic and spherical cases. |
| title | Complete bipartite graphs flexible in the plane |
| topic | Algebraic Geometry Metric Geometry |
| url | https://arxiv.org/abs/2212.11231 |