Pin-Collinear Body-and-Pin Frameworks and the Molecular Conjecture

T-S. Tay and W. Whiteley independently characterized the multigraphs which can be realized as an infinitesimally rigid d-dimensional body-and-hinge framework. In 1984 they jointly conjectured that each graph in this family can be realized as an infinitesimally rigid framework with the additional property that the hinges incident to each body lie in a common hyperplane. This conjecture has become known as the Molecular Conjecture because of its implication for the rigidity of molecules in 3-dimensional space. Whiteley gave a partial solution for the 2-dimensional form of the conjecture in 1989 by showing that it holds for multigraphs G = (V,E) in the family which have the minimum number of edges, i.e. satisfy 2|E| = 3|V | − 3. In this paper, we give a complete solution for the 2-dimensional version of the Molecular Conjecture. Our proof relies on a new formula for the maximum rank of a pin-collinear body-and-pin realization of a multigraph as a 2-dimensional bar-and-joint framework.