Algebraic Properties of the Path Ideal of a Tree

Abstract

The path ideal (of length t ≥ 2) of a directed graph Γ is the monomial ideal, denoted I _t (Γ), whose generators correspond to the directed paths of length t in Γ. We study some of the algebraic properties of I _t (Γ) when Γ is a tree. We first show that I _t (Γ) is the facet ideal of a simplicial tree. As a consequence, the quotient ring R/I _t (Γ) is always sequentially Cohen–Macaulay, and the Betti numbers of R/I _t (Γ) do not depend upon the characteristic of the field. We study the case of the line graph in greater detail at the end of the article. We give an exact formula for the projective dimension of these ideals, and in some cases, we compute their arithmetical rank.

Key Words:

• Arithmetical rank
• Path ideals
• Projective dimension
• Sequentially Cohen–Macaulay
• Simplicial forests

2000 Mathematics Subject Classification:

• 13D02
• 13P10
• 13F55
• 05C99

ACKNOWLEDGMENTS

The second author acknowledges the support of NSERC. This article is partially based upon the M.Sc. project of the first author. We also thank H. T. Hà and the referee for their comments. We would also like to thank Sara Saeedi for pointing out that we needed an extra hypothesis in Lemma 3.6.

Notes

Communicated by R. Villarreal