  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.
