Regularity and projective dimension of the edge ideal of $C_5$-free vertex decomposable graphs

Khosh-Ahang, Fahimeh; Moradi, Somayeh

doi:10.1090/s0002-9939-2014-11906-x

Cited by 44 publications

(24 citation statements)

References 20 publications

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“…(1) G is a sequentially Cohen-Macaulay bipartite graph (see [48]); (2) G is an unmixed bipartite graph (see [36]); (3) G is a very well-covered graph (see [38]); (4) G is a C 5 -free vertex decomposable graph (see [34], the case where G is also C 4 -free was proved in [5]). …”

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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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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“…In this regard, there are many papers devoted to studying this problem (cf. [4], [6], [7], [8], [10], [11], [13] and etc.). Nevertheless most of works are about edge ideals of graphs and so generalizing the gained results on graphs to hypergraphs, to cover all squarefree monomial ideals, makes sense.…”

Section: Introductionmentioning

confidence: 99%

Some interpretations for algebraic invariants of edge ideals of hypergraphs via combinatorial invariants

Moradi

Khosh-Ahang

2020

Communications in Algebra

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The present work is concerned with characterizing some algebraic invariants of edge ideals of hypergraphs. To this aim, firstly, we introduce some kinds of combinatorial invariants similar to matching numbers for hypergraphs. Then we compare them to each other and to previously existing ones. These invariants are used for characterizing or bounding some algebraic invariants of edge ideals of hypergraphs such as graded Betti numbers, projective dimension and Castelnouvo-Mumford regularity.

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“…Computing and finding bounds for the regularity of edge ideals and their powers have been studied by a number of researchers (see for example [1], [2], [3], [4], [5], [8], [10], [15], [16], [17], [18], [20] and [22]).…”

Section: Introductionmentioning

confidence: 99%