Integral closure of powers of edge ideals and their regularity

Let I(G) be the edge ideal of a simple graph G over a field k. We prove thatreg((I(G)(s)) over bar) = 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 that reg I(G)(s) = reg (I(G)(s) over bar) = reg I(G)((s)) = {5 + 2s if char k = 2 4 + 2s if char k &NOTEQUexpressionL; 2for all s >= 1. (c) 2022 Elsevier Inc. All rights reserved.