The nonrepetitive colorings of grids

For a graph $G$, a vertex coloring $f$ is called nonrepetitive if for all $k\in\mathbb N$ and all $P_{2k}=\langle v_1, \cdots, v_k,v_{k+1}, \cdots, v_{2k}\rangle$ (path of $2k$ vertices) in $G$, there must be some $1\le i\le k$ such that $f(v_i)\not=f(v_{k+i})$. We use $\pi(G)$ to denote the minimum number of colors required for $G$ to be nonrepetitively colored. In 1906, Thue proved that $\pi(P_n)\le3$ for all $n$. In this paper, we focus on grids, which are the Cartesian products of paths. We prove that $5\le\pi(P_n\square P_n)\le12$ for sufficiently large $n$, where the previous best lower bound was 4 and upper bound was 16. Moreover, we also discuss nonrepetitive coloring of the Cartesian product of complete graphs.