Let G be a graph and let I be its edge ideal. Our main result shows that the
sets of associated primes of the powers of I form an ascending chain. It is
known that the sets of associated primes of I(i) and intcl(I(i)) stabilize for
large i, where "intcl" denotes integral closure and I(i) denotes the i-th power
of I. We show that for edge ideals their corresponding stable sets are equal.
To show our main result we use a classical result of Berge from matching theory
and certain notions from combinatorial optimization.