  A proper vertex coloring of a graph $G$ is called a star coloring if every
two color classes induce a forest whose each component is a star, which means
there is no bicolored $P_4$ in $G$. In this paper, we show that the Cartesian
product of any two cycles, except $C_3 \square C_3$ and $C_3 \square C_5$, has
a $5$-star coloring.
