A lower bound for depths of powers of edge ideals (Q887944)

Let \(S = k[x_1, \ldots, x_n]\) be a polynomial ring over a field \(k\) in \(n\) indeterminates, and let \(I\) be an \(S\)-ideal generated by quadratic square-free monomials. Considering the monomial generators of \(I\) as the edges of a graph \(G\) on the vertices \(x_1, \ldots, x_n\), the authors get lower bounds for the depth of \(S/I^t\). Specifically, let \(d\) be the diameter, and \(p\) the number of connected components, of \(G\). Then \[ \mathrm{depth}(S/I^t) \geq \left \lceil \frac{d-4t+5}{3} \right \rceil + p -1 \] when \(t \leq 3\) (Theorems 3.1, 4.4 and 4.13). Applying similar arguments, they show that the Stanley depth of \(S/I^t\) is also bounded below by the same function when \(t \leq 3\) (Theorem 4.18).