On the Linear Strand of an Edge Ideal

Roth, Mike; Tuyl, Adam Van

doi:10.1080/00927870601115732

Cited by 27 publications

(16 citation statements)

References 17 publications

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“…Precisely, Theorem 1.1(ii) enables us to give a new proof to the fact that I(G) has property N 2,p for p > 1 if and only if every minimal cycle of G c , the complementary graph of G, has length at least p + 3. Theorem 1.1(ii) also allows us to recover the formulas for the Betti numbers in the linear strand first shown in [21] (cf. Corollary 4.9).…”

Section: Introductionmentioning

confidence: 78%

“…The ideal I(G) is commonly called the edge ideal of G. Edge ideals, which were first introduced by Villarreal [23], are the focus of an ongoing program in algebraic combinatorics. Many authors have been interested in establishing a dictionary between the algebraic invariants of I(G) and the combinatorial data associated to the graph G. References [3,5,[9][10][11]13,14,[17][18][19][21][22][23][24][25][26] form a partial list of references on this topic.…”

Section: Introductionmentioning

confidence: 99%

“…The numbers β i,j (I(G)) are then related to the dimensions of the reduced homology groups of subcomplexes of Δ(G), or as in the case of Eagon and Reiner's formula, the Alexander dual of Δ(G). An examination of the papers [3,11,[17][18][19]21,26] reveals that these formulas provide the basis for most of the known results on the numbers β i,j (I(G)). The exception to this observation is [26] which uses Koszul homology.…”

Section: Introductionmentioning

confidence: 99%

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Splittable ideals and the resolutions of monomial ideals

Harbourne¹,

Tuyl²

2007

Journal of Algebra

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We provide a new combinatorial approach to study the minimal free resolutions of edge ideals, that is, quadratic square-free monomial ideals. With this method we can recover most of the known results on resolutions of edge ideals with fuller generality, and at the same time, obtain new results. Past investigations on the resolutions of edge ideals usually reduced the problem to computing the dimensions of reduced homology or Koszul homology groups. Our approach circumvents the highly nontrivial problem of computing the dimensions of these groups and turns the problem into combinatorial questions about the associated simple graph. We also show that our technique extends successfully to the study of graded Betti numbers of arbitrary square-free monomial ideals viewed as facet ideals of simplicial complexes.

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Section: Introductionmentioning

confidence: 78%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Splittable ideals and the resolutions of monomial ideals

Harbourne¹,

Tuyl²

2007

Journal of Algebra

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“…Many researchers have been interested in using the edge ideal construction to build a dictionary between the fields of graph theory and commutative algebra. For general references, see [26,30,31]; for invariants encoded in the resolution, see [5,7,16,17,18,22,24]; for classes of (sequentially) Cohen-Macaulay graphs, see [9,12,20,29].…”

Section: Introductionmentioning

confidence: 99%

Associated primes of monomial ideals and odd holes in graphs

2010

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Abstract. Let G be a finite simple graph with edge ideal I(G). Let I(G)∨ denote the Alexander dual of I(G). We show that a description of all induced cycles of odd length in G is encoded in the associated primes of (I(G) ∨ ) 2 . This result forms the basis for a method to detect odd induced cycles of a graph via ideal operations, e.g., intersections, products and colon operations. Moreover, we get a simple algebraic criterion for determining whether a graph is perfect. We also show how to determine the existence of odd holes in a graph from the value of the arithmetic degree of (I(G) ∨ ) 2 .

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“…In particular, Villarreal and his coauthors studied Rees rings and Cohen-Macaulay properties of edge ideals [9,10,8]. Hà, Roth and Van Tuyl [4,7], among others, have studied resolutions of edge ideals, finding recursive methods to compute Betti numbers for special classes of graphs. 1 The author acknowledges the support of AARMS.…”

Section: Introductionmentioning

confidence: 99%

The reconstruction conjecture and edge ideals

Dalili

Faridi

Traves

2008

Discrete Mathematics

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Given a simple graph G on n vertices, we prove that it is possible to reconstruct several algebraic properties of the edge ideal from the deck of G, that is, from the collection of subgraphs obtained by removing a vertex from G. These properties include the Krull dimension, the Hilbert function, and all the graded Betti numbers i,j where j < n. We also state many further questions that arise from our study.

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On the Linear Strand of an Edge Ideal

Abstract: Let I(G) be the edge ideal associated to a simple graph G. We study the graded Betti numbers that appear in the linear strand of the minimal free resolution of I(G).2000 Mathematics Subject Classification. 13D40, 13D02, 05C90.

Cited by 27 publications

References 17 publications

Splittable ideals and the resolutions of monomial ideals

Splittable ideals and the resolutions of monomial ideals

Associated primes of monomial ideals and odd holes in graphs

The reconstruction conjecture and edge ideals

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