Global rigidity of complete bipartite graphs

TL;DR

This paper provides a detailed proof that complete bipartite graphs are generically globally rigid in a given dimension, under specific conditions on their partition sizes.

Contribution

It offers a rigorous proof establishing the conditions under which complete bipartite graphs are globally rigid in any dimension.

Findings

Complete bipartite graphs are generically globally rigid in dimension d.

The rigidity depends on the sizes m and n satisfying certain inequalities.

The proof clarifies the conditions for global rigidity in bipartite graphs.

Abstract

This note gives a detailed proof of the following statement. Let $d\in \mathbb{N}$ and $m,n \ge d + 1$, with $m + n \ge \binom{d+2}{2} + 1$. Then the complete bipartite graph $K_{m,n}$ is generically globally rigid in dimension $d$.