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

Abstract

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.

Vertex maps on graphs – Perron–Frobenius theory

“…We do not assume the the underlying map has a certain homotopy type. (More specialized results that take into account the homotopy type of the underlying map are given in [2,3,4]. )…”