Regularity of symbolic powers of complementary edge ideals

Abstract

Let \(G\) be a finite simple graph on \(n\) vertices, and let \(I_c(G)\) be its complementary edge ideal. We determine \(\reg(I_c(G)^{(t)})\) for every \(t\geq1\) in terms of the number \(c(G)\) of nontrivial connected components of \(G\) and the presence of a \(K_2\)-component. Consequently, \(\reg(I_c(G)^{(t)})\leq \reg(I_c(G)^t) \) for all \(t\geq1\), with equality for every \(t\) if and only if either \(c(G)=1\), or \(c(G)=2\) and \(G\) has a \(K_2\)-component. We also characterize Serre's condition \((S_2)\) for \(R/I_c(G)\), and classify the graphs for which \(R/I_c(G)^{(t)}\) is Cohen-Macaulay for every \(t\geq1\).