Edge ideals: algebraic and combinatorial properties

By Susan Morey, Rafael H. Villarreal. 2010

218 citations recorded by the source corpus.

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.

Subfields: Algebra and Number Theory, Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/fb5bca82acc3f1922f08cfd562a0431ee69f9b27

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Citations in the corpus

Regularity 3 in edge ideals associated to bipartite graphs (2012, background): “) associated to them is a well known technique that has been fruitfully exploited in the last decades (see for example the surveys [12] and [15] and their references).”
The stable set of associated prime ideals of a polymatroidal ideal (2011, background): “ls, these questions have been answered quite comprehensively, see [5], [9] and [17]. A nice survey on what is known about the stability of associated prime ideals of powers of edge ideals is given in [18]. Question (3) doesn’t have a positive answer in general, see [12] and [17] for counterexamples.”
Symbolic powers of sums of ideals (2017, background): “Examples of monomial ideals with non-monotone depth functions were rare [3, 19, 27].”
Non-vanishingness of Betti numbers of edge ideals (2011, methodology): “ISHINGNESS OF BETTI NUMBERS 11 which follows from Morey and Villarreal [15, Corollary 3.33] together with Francisco and Van Tuyl [5, Theorem 3.2]; see also Herzog, Hibi, and Zheng [8]. Corollary 5.6 ([15], [5], [8]). Let G be a chordal graph on V. We set S = K[V]. Then bightI(G) = pdS/I(G). In parti”
Algebraic study on Cameron-Walker graphs (2013, background): “Recently, edge ideals of finite simple graphs have been studied by many authors from viewpoints of computational commutative algebra and combinatorics; see [8, 14, 17], and their references.”
Associated primes of powers of edge ideals (2011, unknown): “The second case is the family of graphs with at least one leaf [19], which is now a particular case of our main result.”
The regularity of powers of edge ideals (2014, background): “Bounds on the regularity of edge ideals have been studied by a number of researchers (see [1], [2], [3], [4], [5], [6], [7],[8] [9], [11],[12],[13]).”
A Beginner’s Guide to Edge and Cover Ideals (2013, background): “Finally, if you are interested in learning more about the regularity of edge ideals, the following papers should be part of your reading list: [12, 32, 44, 46, 50, 62, 65, 66, 68].”