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

Abstract

Let $G$ be a simple graph and $I$ its edge ideal. We prove that $$\mathrm{reg}(I^{(s)})=\mathrm{reg}(I^s)$$ 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*}

24 pages, submitted

References in corpus (1)

• Regularity of symbolic powers of edge ideals of Cameron-Walker graphs

Cited by in corpus (1)

• Integral closure of small powers of edge ideals and their regularity