## Abstract

Let R = k[x 1,..,x n], where k is a field. The path ideal (of length t ≥ 2) of a directed graph G is the monomial ideal, denoted by It(G), whose generators correspond to the directed paths of length t in G. Let Cn be an n-cycle. We show that R/It(Cn) is Sr if and only if it is Cohen-Macaulay or n n-t-1 r+3. In addition, we prove that R/It(Cn) is Gorenstein if and only if n = t or 2t + 1. Also, we determine all ordinary and symbolic powers of It(Cn) which are Cohen-Macaulay. Finally, we prove that It(Cn) has a linear resolution if and only if t ≥ n-2.

Original language | English |
---|---|

Pages (from-to) | 7-15 |

Number of pages | 9 |

Journal | Glasgow Mathematical Journal |

Volume | 57 |

Issue number | 1 |

DOIs | |

Publication status | Published - Jan 25 2015 |

Externally published | Yes |

## Keywords

- 05C38
- 05C75
- 13F55
- 2010 Mathematics Subject Classification 13D02

## ASJC Scopus subject areas

- Mathematics(all)

## Fingerprint

Dive into the research topics of 'Gorenstein and S_{r}path ideals of cycles'. Together they form a unique fingerprint.