The nonrepetitive coloring of grids

Abstract

For a graph G, a vertex coloring f is called nonrepetitive if for all $k\in N$ and all $P_{2k}=〈v_1,\cdots ,v_k,v_{k+1},\cdots ,v_{2k}〉$ (path of 2k vertices) in G, there must be some $1\le i\le k$ such that $f\left(v_i\right)\ne f\left(v_{k+i}\right)$.

We use $\pi \left(G\right)$ to denote the minimum number of colors required for G to be nonrepetitively colored.

In 1906, Thue proved that $\pi \left(P_n\right)\le 3$ for all n. In this paper, we focus on grids, which are the Cartesian products of paths. We prove that $5\le \pi (P_n\square P_n)\le 12$ 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.