Edge ideals: algebraic and combinatorial properties
Commutative Algebra
2012-05-23 v3 Combinatorics
Abstract
Let C be a clutter and let I(C) be its edge ideal. This is a survey paper on the algebraic and combinatorial properties of R/I(C) and C, respectively. We give a criterion to estimate the regularity of R/I(C) and apply this criterion to give new proofs of some formulas for the regularity. If C is a clutter and R/I(C) is sequentially Cohen-Macaulay, we present a formula for the regularity of the ideal of vertex covers of C and give a formula for the projective dimension of R/I(C). We also examine the associated primes of powers of edge ideals, and show that for a graph with a leaf, these sets form an ascending chain.
Cite
@article{arxiv.1012.5329,
title = {Edge ideals: algebraic and combinatorial properties},
author = {Susan Morey and Rafael H. Villarreal},
journal= {arXiv preprint arXiv:1012.5329},
year = {2012}
}