On nonrepetitive colorings of paths and cycles

We say that a sequence $a_1\cdots a_{2t}$ of integers is repetitive if $a_i=a_{i+t}$ for every $i\in \{1,\dots ,t\}$. A walk in a graph $G$ is a sequence $v_1\cdots v_r$ of vertices of $G$ in which $v_i{v}_{i+1}\in E\left(G\right)$ for every $i\in \{1,\dots ,r-1\}$. Given a $k$-coloring $c:V\left(G\right)\to \{1,\dots ,k\}$ of $V\left(G\right)$, we say that $c$ is walk-nonrepetitive (resp. stroll-nonrepetitive) if for every $t\in N$ and every walk $v_1\cdots v_{2t}$ the sequence $c\left(v_1\right)\cdots c\left(v_{2t}\right)$ is not repetitive unless $v_i=v_{i+t}$ for every $i\in \{1,\dots ,t\}$ (resp. unless $v_i=v_{i+t}$ for some $i\in \{1,\dots ,t\}$). The walk (resp. stroll) chromatic number $\sigma \left(G\right)$ (resp. $\rho \left(G\right)$) 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 Barát and Wood in 2008: (i) $\sigma \left(C_n\right)=4$ whenever $n\ge 4$ and $n\notin \{5,7\}$; (ii) $\rho \left(P_n\right)=3$ if $3\le n\le 21$ and $\rho \left(P_n\right)=4$ otherwise; and (iii) $\rho \left(C_n\right)=4$, whenever $n\notin \{3,4,6,8\}$, and $\rho \left(C_n\right)=3$ otherwise. In particular, (ii) improves bounds on $n$ obtained by Ochem in 2021 and Tao in 2023.