Abstract. 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 the number of vertices is not a divisor of 2 k then there must be a periodic point with period 2 k . The second is that if the number of vertices is 2 k s 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.
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