Complete bipartite graphs flexible in the plane
Sbornik. Mathematics, Tome 214 (2023) no. 10, pp. 1390-1414
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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.
Bibliography: 11 titles.
Keywords:
complete bipartite graph, flexibility in the plane, algebraic curves.
@article{SM_2023_214_10_a2,
author = {M. D. Kovalev and S. Yu. Orevkov},
title = {Complete bipartite graphs flexible in the plane},
journal = {Sbornik. Mathematics},
pages = {1390--1414},
publisher = {mathdoc},
volume = {214},
number = {10},
year = {2023},
language = {en},
url = {http://geodesic.mathdoc.fr/item/SM_2023_214_10_a2/}
}
M. D. Kovalev; S. Yu. Orevkov. Complete bipartite graphs flexible in the plane. Sbornik. Mathematics, Tome 214 (2023) no. 10, pp. 1390-1414. http://geodesic.mathdoc.fr/item/SM_2023_214_10_a2/