Cohen–Macaulayness for symbolic power ideals of edge ideals

Let S=K[x1,…,xn] be a polynomial ring over a field K. Let I(G)⊆S denote the edge ideal of a graph G. We show that the ℓth symbolic power I(G)(ℓ) is a Cohen–Macaulay ideal (i.e., S/I(G)(ℓ) is Cohen–Macaulay) for some integer ℓ⩾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)(ℓ) are Cohen–Macaulay ideals. Similarly, we characterize graphs G for which S/I(G)(ℓ) has (FLC).As an application, we show that an edge ideal I(G) is complete intersection provided that S/Iℓ(G) is Cohen–Macaulay for some integer ℓ⩾3. This strengthens the main theorem in Crupi et al. (2010) [3].