  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].
