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

Periods of orbits for maps on graphs homotopic to a constant map