The nonrepetitive colorings of grids

TL;DR

This paper investigates the minimum number of colors needed for nonrepetitive vertex coloring of grid graphs, establishing new bounds for large grids and exploring colorings of Cartesian products of complete graphs.

Contribution

It improves bounds on the nonrepetitive chromatic number of grid graphs and extends the discussion to Cartesian products of complete graphs.

Findings

Lower bound for grid graphs: 5 colors

Upper bound for grid graphs: 12 colors

Extended analysis to Cartesian products of complete graphs

Abstract

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.