Generic combinatorial rigidity of periodic frameworks

We give a combinatorial characterization of generic minimal rigidity for planar periodic frameworks. The characterization is a true analogue of the Maxwell–Laman Theorem from rigidity theory: it is stated in terms of a finite combinatorial object and the conditions are checkable by polynomial time combinatorial algorithms.To prove our rigidity theorem we introduce and develop periodic direction networks   and Z2Z2-graded-sparse colored graphs.