Bounds For Invariants of Edge-Rings

Gitler, Isidoro; Valencia, Carlos E.

doi:10.1081/agb-200061033

Cited by 40 publications

(35 citation statements)

References 14 publications

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“…In this case it is well-known that 2ht (I(G)) ≥ |V (G)|. See, e.g., [8]. A graph G is called very well-covered if it is unmixed without isolated vertices and with 2ht (I(G)) = |V (G)|.…”

Section: The Ideal I(g) Is Called the Edge Ideal Of Gmentioning

confidence: 96%

Vertex decomposability and regularity of very well-covered graphs

Mahmoudi

Mousivand

Crupi

et al. 2011

Journal of Pure and Applied Algebra

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a b s t r a c tA graph is called very well-covered if it is unmixed without isolated vertices such that the cardinality of each minimal vertex cover is half the number of vertices. We first prove that a very well-covered graph is Cohen-Macaulay if and only if it is vertex decomposable. Next, we show that the Castelnuovo-Mumford regularity of the quotient ring of the edge ideal of a very well-covered graph is equal to the maximum number of pairwise 3-disjoint edges.

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“…In this case it is well-known that 2ht (I(G)) ≥ |V (G)|. See, e.g., [8]. A graph G is called very well-covered if it is unmixed without isolated vertices and with 2ht (I(G)) = |V (G)|.…”

Section: The Ideal I(g) Is Called the Edge Ideal Of Gmentioning

confidence: 96%

Vertex decomposability and regularity of very well-covered graphs

Mahmoudi

Mousivand

Crupi

et al. 2011

Journal of Pure and Applied Algebra

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“…For an unmixed graph G, it is known (cf. [24]) that 2 ht I(G) ≥ |X(G)|. A graph G is called very well-covered (see [38]) if G is unmixed, has no isolated vertices, and 2 ht I(G) = |X(G)|.…”

Section: Small Regularity and Computing Regularitymentioning

confidence: 99%

Regularity of Squarefree Monomial Ideals

Harbourne

2014

Springer Proceedings in Mathematics &Amp; Statistics

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Abstract. We survey a number of recent studies of the Castelnuovo-Mumford regularity of squarefree monomial ideals. Our focus is on bounds and exact values for the regularity in terms of combinatorial data from associated simplicial complexes and/or hypergraphs.Dedicated to Tony Geramita, a great teacher, colleague and friend. IntroductionCastelnuovo-Mumford regularity (or simply regularity) is an important invariant in commutative algebra and algebraic geometry that governs the computational complexity of ideals, modules and sheaves. Computing or finding bounds for the regularity is a difficult problem. Many simply stated questions and conjectures have been verified only in very special cases. For instance, the Eisenbud-Goto conjecture, which states that the regularity of a projective variety is bounded by the difference between its degree and codimension, has been proved only for arithmetic Cohen-Macaulay varieties, curves and surfaces.The class of squarefree monomial ideals is a classical object in commutative algebra, with strong connections to topology and combinatorics, which continues to inspire much of current research. During the last few decades, advances in computer technology and speed of computation have drawn many researchers' attention toward problems and questions involving this class of ideals. Investigating the regularity of squarefree monomial ideals has evolved to be a highly active research topic in combinatorial commutative algebra.In this paper, we survey a number of recent studies on the regularity of squarefree monomial ideals. Even for this class of ideals, the list of results on regularity is too large to exhaust. Our focus will be on studies that find bounds and/or compute the regularity of squarefree monomial ideals in terms of combinatorial data from associated simplicial complexes and hypergraphs. Our aim is to provide readers with an adequate overall picture of the problems, results and techniques, and to demonstrate similarities and differences between these studies. At the same time, we hope to motivate interested researchers, and especially graduate students, to start working in this area. We shall showcase results in a timeline to exhibit (if possible) trends in developing new and/or more general results from previous ones.The paper is structured as follows. In the next section we collect basic notation and terminology from commutative algebra and combinatorics that will be used in the paper. This section provides readers who are new to the research area the necessary background to start. More advanced readers can skip this section and go directly to Section 3. In Section 1 arXiv:1310.7912v2 [math.AC]

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“…Determining f (n, α, 1) was in fact an old problem of Ore [10] which has been settled recently independently by Christophe et al [3] and by Gitler and Valencia [6]. They proved the following result, where t(n, α) is the size of the Turán graph T (n, α).…”

Section: Introductionmentioning

confidence: 85%

Turán's theorem and k‐connected graphs

Bougard¹,

Joret²

2008

Journal of Graph Theory

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The minimum size of a k-connected graph with given order and stability number is investigated. If no connectivity is required, the answer is given by Turán's Theorem. For connected graphs, the problem has been solved recently independently by Christophe et al., and by Gitler and Valencia. In this paper, we give a short proof of their result and determine the extremal graphs. We settle the case of 2-connected graphs, characterize the corresponding extremal graphs, and also extend a result of Brouwer related to Turán's Theorem. c (Year) John Wiley & Sons, Inc.

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Bounds For Invariants of Edge-Rings

Cited by 40 publications

References 14 publications

Vertex decomposability and regularity of very well-covered graphs

Vertex decomposability and regularity of very well-covered graphs

Regularity of Squarefree Monomial Ideals

Turán's theorem and k‐connected graphs

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