Comparision between regularity of small symbolic powers and ordinary powers of an edge ideal

Let $G$ be a simple graph and $I$ its edge ideal. We prove that $${ \rm reg}(I^{ (s)}) = { \rm reg}(I^s)$$ for $s = 2,3$, where $I^{ (s)}$ is the $s$-th symbolic power of $I$. As a consequence, we prove the following bounds \begin{ align*} { \rm reg} I^{ s} & \le { \rm reg} I + 2s - 2, \text{ for } s = 2,3, { \rm reg} I^{ (s)} & \le { \rm reg} I + 2s - 2,\text{ for } s = 2,3,4. \end{ align*}

Categories: math.AC