Cohen--Macaulaynees for symbolic power ideals of edge ideals

Let $S = K[x_1,..., x_n]$ be a polynomial ring over a field $K$. Let $I(G) \subseteq S$ denote the edge ideal of a graph $G$. We show that the $\ell$th symbolic power $I(G)^{(\ell)}$ is a Cohen-Macaulay ideal (i.e., $S/I(G)^{(\ell)}$ is Cohen-Macaulay) for some integer $\ell \ge 3$ if and only if $G$ is a disjoint union of finitely many complete graphs. When this is the case, all the symbolic powers $I(G)^{(\ell)}$ are Cohen-Macaulay ideals. Similarly, we characterize graphs $G$ for which $S/I(G)^{(\ell)}$ has (FLC). As an application, we show that an edge ideal $I(G)$ is complete intersection provided that $S/I(G)^{\ell}$ is Cohen-Macaulay for some integer $\ell \ge 3$. This strengthens the main theorem in [Effective Cowsik-Nori theorem for edge ideals by M.Crupi, G.Rinaldo, N.Terai, and K.Yoshida, Comm. Alg. 38 (2010), 3347-3357].