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 $s\geq 1$, 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 ${\rm reg}(I(G)^{{s}}$ in terms of the induced matching number of $G$. Moreover, for any graph $G$, we provide sharp upper bounds for ${\rm reg}(I(G)^{{2}})$ and ${\rm reg}(I(G)^{{3}})$.