The minimal Zn-symmetric graphs that are not Zn-spherical

Given a graph G equipped with faithful and fixed-point-free Γ-action (Γ a finite group) we define an orbit minor H of G to be a minor of G for which the deletion and contraction sets are closed under the Γ-action. The orbit minor H inherits a Γ-symmetry from G, and when the contraction set is acyclic the action inherited by H remains faithful and fixed-point free. When G embeds in the sphere and the Γ-action on G extends to a Γ-action on the entire sphere, we say that G is Γ-spherical. In this paper we determine for every odd value of n≥3 the orbit-minor-minimal graphs G with a faithful and free Zn-action that are not Zn-spherical. There are 11 infinite families of such graphs, each of the 11 having exactly one member for each n. For n=3, another such graph is K3,3. The remaining graphs are, essentially, the Cayley graphs for Zn aside from the cycle of length n. The result for n=1 is exactly Wagner's result from 1937 that the minor-minimal graphs that are not embeddable in the sphere are K5 and K3,3.