On the Dynamic Coloring of Cartesian Product Graphs

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 χ2(G). We denote the cartesian product of G and H by G¤H. In this paper, we prove that if G and H are two graphs and δ(G) ≥ 2, then χ2(G¤H) ≤ max(χ2(G), χ(H)). We show that for every two natural numbers m and n, m,n ≥ 2, χ2(Pm¤Pn) = 4. Also, among other results it is shown that if 3|mn, then χ2(Cm¤Cn) = 3 and otherwise χ2(Cm¤Cn) = 4.