Regularity of powers of edge ideals

Let G be a graph and I = I(G) be its edge ideal. In this talk, we review the main results about the regularity of powers of I. Recently, Beyarslam, H`a and Trung proved that reg(I(G)s) ≥ 2s + indmatch(G) − 1, for every s ≥ 1. They also proved that the equality happens for every s ≥ 1, when G is a forest and for every s ≥ 2, when G is a cycle. In this talk, we determine a new class of graphs for which the equality holds. More explicit, we show that if G is a whiskered cycle graph, then reg(Is) = 2s + ⌈n−1/2 ⌉ − 1, for every integer s ≥ 1.