Depth of powers of edge ideals of cycles and trees

Abstract

Let $I$ be the edge ideal of a cycle of length $n\ge 5$ over a polynomial ring $S=\mathrm{k}[x_1,\dots ,x_n]$. We prove that for $2\le t<\lceil (n+1)/2\rceil$, $$\mathrm{depth}(S/I^t)=\lceil \frac{n-t+1}{3}\rceil .$$ When $G=T_{\mathbf{a}}$ is a starlike tree which is the join of $k$ paths of length $a_1,\dots ,a_k$ at a common root $1$, we give a formula for the depth of powers of $I(T_{\mathbf{a}})$.