  A coloring $c$ of the vertices of a graph $G$ is nonrepetitive if there
exists no path $v_1v_2\ldots v_{2l}$ for which $c(v_i)=c(v_{l+i})$ for all
$1\le i\le l$. Given graphs $G$ and $H$ with $|V(H)|=k$, the lexicographic
product $G[H]$ is the graph obtained by substituting every vertex of $G$ by a
copy of $H$, and every edge of $G$ by a copy of $K_{k,k}$. %Our main results
are the following. We prove that for a sufficiently long path $P$, a
nonrepetitive coloring of $P[K_k]$ needs at least $3k+\lfloor k/2\rfloor$
colors. If $k>2$ then we need exactly $2k+1$ colors to nonrepetitively color
$P[E_k]$, where $E_k$ is the empty graph on $k$ vertices. If we further require
that every copy of $E_k$ be rainbow-colored and the path $P$ is sufficiently
long, then the smallest number of colors needed for $P[E_k]$ is at least $3k+1$
and at most $3k+\lceil k/2\rceil$. Finally, we define fractional nonrepetitive
colorings of graphs and consider the connections between this notion and the
above results.
