## Abstract

Let S=K[x_{1},..., x_{n}] 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/I(G)^{ℓ} is Cohen-Macaulay for some integer ℓ≥3. This strengthens the main theorem in Crupi et al. (2010) [3].

Original language | English |
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Pages (from-to) | 1-22 |

Number of pages | 22 |

Journal | Journal of Algebra |

Volume | 347 |

Issue number | 1 |

DOIs | |

Publication status | Published - Dec 1 2011 |

Externally published | Yes |

## Keywords

- Cohen-Macaulay
- Complete intersection
- Edge ideal
- FLC
- Polarization
- Simplicial complex
- Symbolic powers

## ASJC Scopus subject areas

- Algebra and Number Theory