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

Sharkovsky's Theorem and one-dimensional combinatorial dynamics

“…Similar types of arguments can also be used for vertex maps from graphs to themselves. In the case when the map is homotopic to the constant map, it is shown in [9] that one obtains exactly the same ordering as in Theorem 8, and in [8] it is shown that the same result holds for maps on graphs that are homotopic to the identity map and that flip an edge (i.e., there is an edge in the graph that gets mapped onto itself with its orientation reversed).…”

“…These arguments about periodic orbits can often be reduced to elementary arguments involving the traces of matrices, and these are accessible to undergraduates. Indeed, [9] was written as part of an REU project at Fairfield University.…”

“…We now give an example to show that the results in the previous section do not extend to these cases. Consider the map on the tree shown in Figure 10 that maps the vertices according to the permutation (1,2,3,4,5,6,7,8,9). The Markov Graph is shown in Figure 11.…”