On Small Separations in Cayley and Vertex Transitive Graphs

DeVos and Mohar proved a rough structure theorem about small separations in vertex-transitive graphs [5, 6]. By using a new version of Varopolous isoperimetric inequality, we give an improvement of their lower bound on the expansion in the case of infinite vertex-transitive graphs. Specifically, let (Formula presented.) be a locally finite connected graph such that there is a group G acting discretely and transitively on X. If (Formula presented.) is nonempty and finite such that (Formula presented.) is connected and (Formula presented.), then X has a ring-like structure. Moreover, we give a similar asymptotic result under the assumption that X is an infinite vertex transitive graph. In the setting of finite groups, we use local expansion to show the existence of a nontrivial cyclic subgroup with an effectively bounded index. Finally, we prove that for any (Formula presented.) there is a graph T and a subgraph A of T, such that (Formula presented.). © 2015 Wiley Periodicals, Inc