A proof of the Molecular Conjecture

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
$\left({d+1 \choose 2}-1\right)G$ contains ${d+1\choose 2}$
edge-disjoint spanning trees, where $\left({d+1 \choose 2}-1\right)G$
is the graph obtained from G
by replacing each edge by $\left({d+1\choose 2}-1\right)$ 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 bofy-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.
Recently we proved this long standing conjecture affirmatively
for general dimension.
Also, as a corollary, we obtain a combinatorial characterization of
the 3-dimensional bar-and-joint rigidity matroid
of the square of a graph.
This is a joint work with Shin-ichi Tanigawa.