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Integral closure of small powers of edge ideals and their regularity (2109.09268v2)

Published 20 Sep 2021 in math.AC and math.CO

Abstract: Let $I(G)$ be the edge ideal of a simple graph $G$ over a field k. We prove that $${\rm reg}(\overline {I(G)s}) = {\rm reg}(I(G)s),$$ for all $s \le 4$. Furthermore, we provide an example of a graph $G$ such that $${\rm reg} I(G)s = {\rm reg} \overline{I(G)s} = {\rm reg} I(G){(s)} = \begin{cases} 5 + 2s & \text{ if char k} = 2 \ 4 + 2s & \text{ if char k} \neq 2, \end{cases}$$ for all $s \ge 1.$

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