Algebraic properties of the path ideal of a tree
Commutative Algebra
2009-10-06 v3 Combinatorics
Abstract
The path ideal (of length t >=2) of a graph G is the monomial ideal, denoted I_t(G), whose generators correspond to the directed paths of length t in G. We study some of the algebraic properties of I_t(G) when G is a tree. We first show that I_t(G) is the facet ideal of a simplicial tree. As a consequence, the quotient ring R/I_t(G) is always sequentially Cohen-Macaulay, and the Betti numbers of R/I_t(G) 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 paper.
Keywords
Cite
@article{arxiv.0902.0902,
title = {Algebraic properties of the path ideal of a tree},
author = {Jing Jane He and Adam Van Tuyl},
journal= {arXiv preprint arXiv:0902.0902},
year = {2009}
}
Comments
Added some additional hypotheses to old Theorem 3.6; now Lemma 3.6 and Theorem 3.7