On nonrepetitive colorings of paths and cycles

Abstract: We say that a sequence $a_1 \cdots a_{2t}$ of integers is repetitive if $a_i = a_{i+t}$ for every $i\in{1,\ldots,t}$. A walk in a graph $G$ is a sequence $v_1 \cdots v_r$ of vertices of $G$ in which $v_iv_{i+1}\in E(G)$ for every $i\in{1,\ldots,r-1}$. Given a $k$-coloring $c\colon V(G)\to{1,\ldots,k}$ of $V(G)$, we say that $c$ is walk-nonrepetitive (resp. stroll-nonrepetitive) if for every $t\in\mathbb{N}$ and every walk $v_1\cdots v_{2t}$ the sequence $c(v_1) \cdots c(v_{2t})$ is not repetitive unless $v_i = v_{i+t}$ for every $i\in{1,\ldots,t}$ (resp. unless $v_i = v_{i+t}$ for some $i\in{1,\ldots,t}$). The walk (resp. stroll) chromatic number $\sigma(G)$ (resp. $\rho(G)$) of $G$ is the minimum $k$ for which $G$ has a walk-nonrepetitive (resp. stroll-nonrepetitive) $k$-coloring. Let $C_n$ and $P_n$ denote, respectively, the cycle and the path with $n$ vertices. In this paper we present three results that answer questions posed by Bar\'at and Wood in 2008: (i) $\sigma(C_n) = 4$ whenever $n\geq 4$ and $n \notin{5,7}$; (ii) $\rho(P_n) = 3$ if $3\leq n\leq 21$ and $\rho(P_n) = 4$ otherwise; and (iii) $\rho(C_n) = 4$, whenever $n \notin{3,4,6,8}$, and $\rho(C_n) = 3$ otherwise. In particular, (ii) improves bounds on $n$ obtained by Tao in 2023.