Complete bipartite graphs flexible in the plane

TL;DR

This paper investigates the flexibility of complete bipartite graphs $K_{m,n}$ as planar linkages across Euclidean, spherical, and hyperbolic geometries, providing comprehensive classifications and proofs without heavy computations.

Contribution

It offers a complete classification of the flexibility of all $K_{m,n}$ graphs in various geometries with self-contained proofs, extending previous partial results.

Findings

Classified all $K_{m,n}$ graphs for flexibility in Euclidean, spherical, and hyperbolic planes.

Provided new, simplified proofs applicable across different geometries.

Extended known results to all complete bipartite graphs, resolving longstanding questions.

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.