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 $s \ge 1.$
Categories: math.AC math.CO