  In this paper, some results concerning the colorings of graph powers are
presented. The notion of helical graphs is introduced. We show that such graphs
are hom-universal with respect to high odd-girth graphs whose $(2t+1)$st power
is bounded by a Kneser graph. Also, we consider the problem of existence of
homomorphism to odd cycles. We prove that such homomorphism to a $(2k+1)$-cycle
exists if and only if the chromatic number of the $(2k+1)$st power of $S_2(G)$
is less than or equal to 3, where $S_2(G)$ is the 2-subdivision of $G$. We also
consider Ne\v{s}et\v{r}il's Pentagon problem. This problem is about the
existence of high girth cubic graphs which are not homomorphic to the cycle of
size five. Several problems which are closely related to Ne\v{s}et\v{r}il's
problem are introduced and their relations are presented.
