A Proof of the Molecular Conjecture

TL;DR

This paper proves the long-standing Molecular Conjecture, establishing a combinatorial characterization for the rigidity of body-and-hinge frameworks in any dimension, linking geometric properties to graph-theoretic conditions.

Contribution

It provides a proof of the Molecular Conjecture for all dimensions, extending the 2D case to higher dimensions and confirming the combinatorial criteria for rigidity.

Findings

The conjecture holds true in all dimensions.

Hinge-coplanar property characterizes rigidity.

Graph-theoretic conditions are necessary and sufficient.

Abstract

A $d$-dimensional body-and-hinge framework is a structure consisting of rigid bodies connected by hinges in $d$-dimensional space. The generic infinitesimal rigidity of a body-and-hinge framework has been characterized in terms of the underlying multigraph independently by Tay and Whiteley as follows: A multigraph $G$ can be realized as an infinitesimally rigid body-and-hinge framework by mapping each vertex to a body and each edge to a hinge if and only if ({d+1 \choose 2}-1)G$ contains ${d+1\choose 2}$ edge-disjoint spanning trees, where $({d+1 \choose 2}-1)G$ is the graph obtained from $G$ by replacing each edge by $({d+1\choose 2}-1)$ parallel edges. In 1984 they jointly posed a question about whether their combinatorial characterization can be further applied to a nongeneric case. Specifically, they conjectured that $G$ can be realized as an infinitesimally rigid body-and-hinge framework if and only if $G$ can be realized as that with the additional ``hinge-coplanar'' property, i.e., all the hinges incident to each body are contained in a common hyperplane. This conjecture is called the Molecular Conjecture due to the equivalence between the infinitesimal rigidity of ``hinge-coplanar'' body-and-hinge frameworks and that of bar-and-joint frameworks derived from molecules in 3-dimension. In 2-dimensional case this conjecture has been proved by Jackson and Jord{\'a}n in 2006. In this paper we prove this long standing conjecture affirmatively for general dimension.