On the rigidity of molecular graphs

The rigidity of squares of graphs in three-space has important applications to the study of flexibility in molecules. The Molecular Conjecture, posed in 1984 by T-S. Tay and W. Whiteley, states that the square G of a graph G of minimum degree at least two is rigid if and only if G has six spanning trees which cover each edge of G at most five times. We give a lower bound on the degrees of freedom of G in terms of forest covers of G. This provides a self-contained proof that the existence of the above six spanning trees is a necessary condition for the rigidity of G. In addition, we prove that the truth of the Molecular Conjecture would imply that our lower bound is tight, and would also imply that a conjecture of Jacobs on ‘independent’ squares is valid.