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The Minimal Z_n-Symmetric Graphs That Are Not Z_n-Planar
by
Lowell Abrams
The George Washington University
Coauthors: Daniel Slilaty (Wright State University)

Given a graph G equipped with 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 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. We determine for every odd value of n ≥ 3 the orbit-minor-minimal graphs G with a free Z_n-action that are not Z_n-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 K_{3, 3}. The remaining graphs are, essentially, the Cayley graphs for Z_n 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 K_{3, 3}.

Date received: March 2, 2015

Copyright © 2015 by the author(s). The author(s) of this work and the organizers of the conference have granted their consent to include this abstract in Topology Atlas. Document # cbkp-20.