Depth and Stanley depth of the edge ideals of square paths and square cycles

ABSTRACT

In this paper, we compute depth and Stanley depth for the quotient ring of the edge ideal associated to a square path on n vertices. We also compute depth and Stanley depth for the quotient ring of the edge ideal associated to a square cycle on n vertices, when n≡0,3,4( mod 5), and give tight bounds when n≡1,2( mod 5). We also prove a conjecture of Herzog presented in [5], for the edge ideals of square paths and square cycles.

KEYWORDS:

• Depth
• edge ideal
• monomial ideal
• square cycle
• square path
• Stanley decomposition
• Stanley depth

2000 MATHEMATICS SUBJECT CLASSIFICATION:

• Primary: 13C15
• Secondary: 13F20, 05C38, 05E99

Acknowledgment

The authors would like to thank the referee for a careful reading of the paper and for valuable comments. The authors would like to thank the referee for a careful reading of the paper and for valuable comments. This research is partially supported by HEC Pakistan.