Let $G$ be an undirected odd cycle. Let $f$ be a proper 3-coloring of $G$. If $w=v\_1v\_2...v\_k$ is a walk on $k$ vertices of $G$, let $f(w)=f(v\_1)f(v\_2)...f(v\_k)$. Let $W\_k=\{f(w)|w$ is a walk on $k$ vertices in $G\}$. Let $|W\_k|$ be the cardinality of $W\_k$. 


Is it true that $\lim\_{k \to \infty} \frac{\log |W\_k|}{k}=\log 2$ regardless of the order of $G$ and the choice of $f$?


Thank you.



Counting walks on proper colorings of odd cycles
Certainly not. Take the coloring that is alternating black and white, except for a single red. 


