Primitiveness conditions for systems of two graphs

Some sufficient conditions for primitiveness of two $n$-vertex digraphs system are obtained in the case when there are no acyclic vertices in one of this two graphs, particularly when it contains a Hamiltonian cycle. Also, an exponent estimate for the two digraphs system is obtained in terms of the exponent of their product. The results can be used for evaluation of the mixing properties of iterative functions based on the transformation branching into two given transformations.