Bounding the socles of powers of squarefree monomial ideals

Let $S=K[x_1,ldots,x_n]$ be the polynomial ring in $n$ variables over a field $K$ and $Isubset S$ a squarefree monomial ideal. In the present paper we are interested in the monomials $u in S$ belonging to the socle $Soc(S/I^{k})$ of $S/I^{k}$, i.e., $u otin I^{k}$ and $ux_{i} in I^{k}$ for $1 leq i leq n$. We prove that if a monomial $x_1^{a_1}cdots x_n^{a_n}$ belongs to $Soc(S/I^{k})$, then $a_ileq k-1$ for all $1 leq i leq n$. 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 $[n] = {1, ldots, n}$ for which $depth S/(I_{G})^{2}=0$, where $I_{G}$ is the edge ideal of $G$.