The exponent of Cartesian product of cycles

A digraph DD is primitive if for each pair of vertices v,wv,w of DD, there is a positive integer kk such that there is a directed walk of length kk from vv to ww. The minimum of such kk is the exponent of DD. In this paper, we show that for a primitive graph GG and a strongly connected bipartite digraph DD, the exponent of the Cartesian product G×DG×D is equal to the addition of the exponent of GG and the diameter of DD. Finally, we find the exponents of Cartesian products of cycles.