Saturation and associated primes of powers of edge ideals

For the edge ideal I   of an arbitrary simple graph Γ we describe the monomials of the saturation of ItIt in terms of (vertex) weighted graphs associated with the monomials. This description allows us to characterize the embedded associated primes of ItIt as covers of Γ which contain certain types of subgraphs of Γ. As an application, we completely classify the associated primes of I2I2 and I3I3 in terms of Γ.