# On the existence of paradoxical motions of generically rigid graphs on the sphere

## TL;DR

This paper explores the conditions under which certain graphs on the sphere exhibit paradoxical motions, providing a combinatorial characterization and classifying specific flexible configurations.

## Contribution

It offers a novel interpretation of spherical graph realizations via moduli spaces and characterizes flexible graphs through edge colorings and length relations.

## Key findings

- Characterization of flexible spherical graphs via edge colorings
- Necessary relations for edge lengths to ensure flexibility
- Complete classification of motions for the K_{3,3} bipartite graph

## Abstract

We interpret realizations of a graph on the sphere up to rotations as elements of a moduli space of curves of genus zero. We focus on those graphs that admit an assignment of edge lengths on the sphere resulting in a flexible object. Our interpretation of realizations allows us to provide a combinatorial characterization of these graphs in terms of the existence of particular colorings of the edges. Moreover, we determine necessary relations for flexibility between the spherical lengths of the edges. We conclude by classifying all possible motions on the sphere of the complete bipartite graph with $3+3$ vertices where no two vertices coincide or are antipodal.