On Some Graphic Aspects of Addition Theorems

During the last fifty years, Davenport, Mann, Kempermann and many others proved some inequalities relating the cardinality of the sum of two sets in a group to the cardinalities of the original sets. This subject is known as Additive Group Theory.Almost all these questions have a natural interpretation as statements about the connectivity of Cayley directed graphs.The Cauchy-Davenport theorem is equivalent to the fact that a Cayley graph with prime order has connectivity equal to the outdegree. Chowla’s Theorem says that the same property holds for Cayley graphs on ℤ_n defined by subsets of ℤ_n*.We was able to prove a common generalization to several addition theorems with the following graphic interpretation.Let G be a finite Abelian group and B⊂G\0. There exists a nonnull subgroup H of G such that any cut separating two elements of H has cardinality at least $$\left| B \right|$$. In other words the local connectivities inside H are optimal.This subgroup exists also when the group is not Abelian and B=B−1.