On the index of depth stability of symbolic powers of cover ideals of graphs (Q6650704)

Let \(G\) be a graph with \(n\) vertices and \(S=K[x_{1},\dots,x_n]\) be the polynomial ring in \(n\) variables over a field \(K\). The paper provides a combinatorial upper bound for the index of depth stability of symbolic powers of \(J(G)\), the cover ideal of the graph \(G\). The index of depth stability of powers of an ideal \(I\) of \(S\) here is the smallest integer \(k\geq 1\) such that \(\operatorname{depth}(S/I^m)=\lim_{t\to \infty}\operatorname{depth}(S/I^t)\) for all \(m\geq k\). This is therefore helpful for determining the limit of the sequence \(\{\operatorname{depth}(S/I^k)\}_{k\in\mathbb{N}}\) which is itself of great interest. Also, same problems are considered and explained for symbolic powers \(I^{(k)}\) of an ideal and the paper computes the depth of symbolic powers of cover ideals of fully clique-whiskered graphs. It is already known ([\textit{L. T. Hoa} et al., J. Algebra 473, 307--323 (2017; Zbl 1358.13020)]) that \(\operatorname{dstab}(J(G)\leq 2\operatorname{ord-match}(G)-1\) and the present paper provides an improvement of this bound (Theorem 3.1). Here \(\operatorname{ord-match}(G)\) denotes the order matching of a graph \(G\) (Definition 2.2). The authors also determine a class of graphs \(G\) with the property that the Castelnuovo-Mumford regularity of \(S/I(G)\) is equal to the induced matching number of \(G\) (another invariant introduced in Section 2, page 368 of the paper). Recall that if \(M\) is a finitely generated graded \(S\)-Module, then Castelnuovo-Mumford regularity (or simply, regularity) of M is defined as \(\max\{j-i\mid\mathrm{Tor}_i(K,M)_j\neq 0\}\).