Complete bipartite graphs flexible in the plane

Abstract

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.

21 pages, 6 figures. Changes with respect to v1:: 1). The hyperbolic and spherical cases are included. 2). References [3] and [5] are added

References in corpus (2)

• On the existence of paradoxical motions of generically rigid graphs on the sphere
• On the Classification of Motions of Paradoxically Movable Graphs