  The paper proves two theorems concerning the set of periods of periodic
orbits for maps of graphs that are homotopic to the constant map and such that
the vertices form a periodic orbit. The first result is that if $v$ is not a
divisor of $2^k$ then there must be a periodic point with period $2^k$. The
second is that if $v=2^ks$ for odd $s>1$, then for all $r>s$ there exists a
periodic point of minimum period $2^k r$. These results are then compared to
the Sharkovsky ordering of the positive integers.
  (The final version of this paper will appear in the Journal of Difference
Equations and Applications.)
