Integral closure of powers of edge ideals and their regularity

Abstract

Let I(G) be the edge ideal of a simple graph G over a field k. We prove thatreg(I(G)s‾)=reg(I(G)s), for all s≤4. For Stanley-Reisner ideals of one-dimensional simplicial complexes, we compute the regularity of integral closure of all powers. We further provide an example of a graph G such thatregI(G)s=regI(G)s‾=regI(G)(s)={5+2sif chark=24+2sif chark≠2, for all s≥1.