On the Regularity of squarefree part of symbolic powers of edge ideals

Abstract

Assume that $G$ is a graph with edge ideal $I(G)$. For every integer $s\ge 1$, we denote the squarefree part of the $s$-th symbolic power of $I(G)$ by $I(G{)}^{\{s\}}$. We determine an upper bound for the regularity of $I(G{)}^{\{s\}}$ when $G$ is a chordal graph. If $G$ is a Cameron-Walker graphs, we compute $\mathrm{reg}(I(G{)}^{\{s\}}$ in terms of the induced matching number of $G$. Moreover, for any graph $G$, we provide sharp upper bounds for $\mathrm{reg}(I(G{)}^{\{2\}})$ and $\mathrm{reg}(I(G{)}^{\{3\}})$.