Edge ideals: algebraic and combinatorial properties (Q639477)

A clutter \(\mathcal C\) is a finite set \(X\) with a family \(E\) of subsets of \(X\) such that if \(f_1,f_2\in E\) then \(f_1\not\subset f_2\). Let \(\mathcal C\) be a clutter and \(I(\mathcal C)\subset R=K[x_1,\dots,x_n]\) its edge ideal. \(I(\mathcal C)\) is the ideal of \(R\) generated by all monomials \(x_e=\prod_{x_i\in e} x_i\) such that \(e\in E(\mathcal C)\). This is a survey paper on the algebraic and combinatorial properties of \(R/I(\mathcal C)\) and \(\mathcal C\). The paper includes some new proofs of known results and some new results.NEWLINENEWLINESection 2 of the paper is devoted to the algebraic and combinatorial properties of edge ideals and focuses on the sequentially Cohen-Macaulay property.NEWLINENEWLINESection 3 considers some important invariants of edge ideals: regularity, projective dimension and depth. This section contains most of the main new results of the paper. Theorem 3.14 gives a criterion to estimate the regularity of edge ideals, and Theorem 3.31 gives a formula for the regularity of the ideal of vertex covers of \(\mathcal C\) in case \(R/I(\mathcal C)\) is sequentially Cohen-Macaulay.NEWLINENEWLINEIn Section 4 the authors study the stability of the associated primes of edge ideals and the connection between torsion-freeness and some combinatorial problems. Proposition 4.23 in this section gives a new class of monomial ideals for which the sets of associated primes of powers are known to form ascending chains.NEWLINENEWLINEThe paper contains a large and complete list of references for the subject, which is of current interest in commutative algebra and combinatorics.NEWLINENEWLINEFor the entire collection see [Zbl 1237.13005].