Equality of regularity for powers and symbolic powers of complementary edge ideals

Determine whether, for every finite simple graph G on the vertex set [n], the equality reg I_c(G)^k = reg I_c(G)^{(k)} holds for all integers k \ge 1, where S = K[x_1,\ldots,x_n], I_c(G) = ((x_1\cdots x_n)/(x_i x_j) : \{i,j\} \in E(G)) is the complementary edge ideal, and I_c(G)^{(k)} denotes the k-th symbolic power of I_c(G).

Background

The authors compute explicit regularity functions for powers of complementary edge ideals I_c(G) and prove additional structural properties (such as Betti splittings and field-independence of graded Betti numbers). They also establish that depth(S/I_c(G)^k) is non-increasing in k.

In contrast, while complementary cover ideals J_c(G) have known equality reg J_c(G)^{(k)} = reg J_c(G)^k for all k \ge 1 from earlier work, it remains unresolved whether an analogous equality reg I_c(G)^{(k)} = reg I_c(G)^k holds for complementary edge ideals. The authors explicitly state this as an open question.