Let G and H be two graphs. A proper vertex coloring of G is called a dynamic coloring, if for every vertex v with degree at least 2, the neighbors of v receive at least two different colors. The smallest integer k such that G has a dynamic coloring with k colors denoted by chi(2)(G). We denote the cartesian product of G and H by G square H. In this paper, we prove that if G and H are two graphs and delta(G) >= 2, then chi(2)(G square H) <= max(chi(2)(G), chi(H)). We show that for every two natural numbers m and n, m, n >= 2, chi(2) (P-m square P-n) = 4. Also, among other results it is shown that if 3 vertical bar mn, then chi(2)(C-m square C-n) = 3 and otherwise chi(2) (C-m square C-n) = 4.
