New lower bounds on the non-repetitive chromatic number of some graphs

A graph \( G \) is said to be (vertex) non-repetitively colored if no simple path in \( G \) has a sequence of vertex colors that forms a repetition. Formally, a coloring \( c: V(G) \to \{1, 2, \dots, k\} \) is non-repetitive if, for every path \(\langle v_1, v_2, \dots, v_{2m} \rangle\) in \( G \), the sequence of colors \( c(v_1), c(v_2), \dots, c(v_{2m}) \) is not of the form \( ww \), where \( w \) is a sequence of \( m \) colors. The minimum number of colors required for such a coloring is called the \emph{non-repetitive chromatic number} of \(G\), denoted by \(π(G)\). In this paper, we primarily prove that \(π(P \square P) \ge 6\) and \(π(P \boxtimes P) \ge 9\), where \( P \square P \) and \( P \boxtimes P \) are the Cartesian product and the strong product of two infinite paths, respectively. This improves upon the previous best lower bounds.