New lower bounds on the non-repetitive chromatic number of some graphs
Tianyi Tao, Junchi Zhang, Wentao Zhang, Alex Toole
TL;DR
The paper addresses lower bounds for the non-repetitive chromatic number $\pi(G)$ of grid-like graphs, focusing on the infinite grid $P \square P$ and the infinite strong product $P \boxtimes P$. It develops an algorithmic framework that extends Toole's method by incrementally constructing a subgraph $H$, enumerating non-repetitive $c$-colorings with symmetry reduction, and checking for repetitive paths via a designated vertex, then applies this approach to $P \square P$, $P \boxtimes P$, and the triangular tiling $T_3$. The main contributions are the new lower bounds $\pi(P \square P) \ge 6$ and $\pi(P \boxtimes P) \ge 9$, supported by computational data and an open-source implementation, as well as a lower bound $\pi(T_3) \ge 9$ stemming from the containment $T_3 \subseteq P \boxtimes P$. The work sheds light on non-repetitive colorings in grid-like graphs, highlights the role of subgraph selection and symmetry in computational approaches, and discusses limitations and extensions to dense graph families such as $K_n \square K_n$.
Abstract
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.