Associated primes of powers of edge ideals

Abstract: 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.