Depth and Stanley depth of symbolic powers of cover ideals of graphs

Let G be a graph with n vertices and let S=K[x1,…,xn] be the polynomial ring in n variables over a field K. Assume that J(G) is the cover ideal of G and J(G)(k) is its k-th symbolic power. We prove that the sequences {sdepth(S/J(G)(k))}k=1∞ and {sdepth(J(G)(k))}k=1∞ are non-increasing and hence convergent. Suppose that νo(G) denotes the ordered matching number of G. We show that for every integer k≥2νo(G)−1, the modules J(G)(k) and S/J(G)(k) satisfy the Stanley's inequality. We also provide an alternative proof for [9, Theorem 3.4] which states that depth(S/J(G)(k))=n−νo(G)−1, for every integer k≥2νo(G)−1.