The Dress conjectures on rank in the 3-dimensional rigidity matroid

A. Dress has made two conjectures concerning the rank function of the 3-dimensional rigidity matroid. The first would give a min-max formula for this rank function and hence a good characterization for independence. We show that the first conjecture is false for all graphs with at least 56 vertices. On the other, hand we show that the second conjecture and a modified form of the first conjecture are true for certain families of graphs of maximum degree at most five.