The generic rank of body-bar-and-hinge frameworks

Tay [6] characterized the multigraphs which can be realized as infinitesimally rigid d-dimensional body-and-bar frameworks. Subsequently, Tay [7] and Whiteley [11] independently characterized the multigraphs which can be realized as infinitesimally rigid d-dimensional body-and-hinge frameworks. We adapt Whiteley’s proof technique to characterize the multigraphs which can be realized as infinitesimally rigid d-dimensional body-bar-and-hinge frameworks. More importantly, we obtain a sufficient condition for a multigraph to be realized as an infinitesimally rigid d-dimensional body-and-hinge framework in which all hinges lie in the same hyperplane. This result is related to a longstanding conjecture of Tay and Whiteley [8] which would characterize when a multigraph can be realized as an infinitesimally rigid d-dimensional body-andhinge framework in which all the hinges incident to each body lie in a common hyperplane. As a corollary we deduce that if a graph G has two spanning trees which use each edge of G at most twice, then its square can be realized as an infinitesimally rigid 3-dimensional bar-and-joint framework.