Bounding the socles of powers of squarefree monomial ideals

Let $S=K[x_1,\dots ,x_n]$ be the polynomial ring in $n$ variables over a field $K$ and $I\subset S$ a squarefree monomial ideal. In the present paper we are interested in the monomials $u\in S$ belonging to the socle $\text{Soc}(S\mathrm{/}{I}^k)$ of $S\mathrm{/}{I}^k$, i.e., $u\in ̸I^k$ and $u{x}_i\in I^k$ for $1\le i\le n$. We prove that if a monomial $x_1^{a_1}\cdots x_n^{a_n}$ belongs to $\text{Soc}(S\mathrm{/}{I}^k)$, then $a_i\le k-1$ for all $1\le i\le n$. We then discuss squarefree monomial ideals $I\subset S$ for which $x_{[n]}^{k-1} \in \Soc(S/I^{k})$,where$x_{[n]} = x_{1}x_{2}\cdots x_{n}$. Furthermore, we give a combinatorial characterization of finite graphs $G$ on $[n] = {1, \ldots, n}$forwhich$\depth S/(I_{G})^{2}=0$,where$I_{G}$ is the edge ideal of $G$.