Flows, Connectivity and Matching

Abstract

Given a collection of boys and girls, when can all the girls find husbands that they know? For a subgroup H of a finite group G, are there group elements g ₁,…, g _n such that {g ₁H,…, g _nH} is the collection of left cosets and {Hg ₁,…, Hg _n} is the collection of right cosets of H? Given sets A ₁,…, A _m, are there distinct elements a ₁ ∈ A ₁,…, a _m ∈ A _m?