Monotonicity of depth for powers and symbolic powers of complementary cover ideals

Determine whether, for every finite simple graph G on the vertex set [n] and the complementary cover ideal J_c(G) = I_c(G)^\vee = \bigcap_{\{i,j\}\in E(G)} (x_i, x_j, x_{k\ne i,j}), the sequences k \mapsto depth(S/J_c(G)^k) and k \mapsto depth(S/J_c(G)^{(k)}) are non-increasing for all integers k \ge 1, where S = K[x_1,\ldots,x_n] and J_c(G)^{(k)} denotes the k-th symbolic power of J_c(G).

Background

The paper introduces complementary edge ideals I_c(G) and their Alexander duals, the complementary cover ideals J_c(G), associated to finite simple graphs G. Among the main results, the authors compute explicit formulas for reg I_c(G)^k and show that the depth function k \mapsto depth(S/I_c(G)^k) is non-increasing.

For complementary cover ideals J_c(G), prior work by Hoa–Trung and Minh–Vu established explicit regularity formulas for powers and symbolic powers and proved reg J_c(G)^{(k)} = reg J_c(G)^k for all k \ge 1. Motivated by the non-increasing depth behavior proved for I_c(G), the authors pose the open question of whether analogous non-increasing behavior holds for the depth functions of the ordinary and symbolic powers of J_c(G).