Depth stability of edge ideals
  Let $G$ be a connected finite simple graph and let $I_G$ be the edge ideal of $G$. The smallest number $k$ for which $\depth S/I_G^k$ stabilizes is denoted by $\dstab(I_G)$. We show that $\dstab(I_G)<\ell(I_G)$ where $\ell(I_G)$ denotes the analytic spread of $I$. For trees we give a stronger upper bound for $\dstab(I_G)$. We also show for any two integers $1\leq a<b$ there exists a tree for which $\dstab(I_G)=a$ and $\ell(I_G)=b$. 
J\"urgen Herzog, Takayuki Hibi
math.AC