Matchings, coverings, and Castelnuovo-Mumford regularity

Woodroofe, Russ

doi:10.1216/jca-2014-6-2-287

Cited by 121 publications

(105 citation statements)

References 37 publications

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“…A number of generalizations for Theorem 4.4 has been developed. Woodroofe [51] was the first to observe that giving a matching in a graph G is a simple way to "cover" the edges of G by subgraphs. One can also consider giving a matching in G as a special way to "pack" edges in G. These ideas have been extended to give better bounds for graphs and to obtain bounds for hypergraphs in general.…”

Section: Combinatorial Bounds For 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: Combinatorial Bounds For Regularitymentioning

confidence: 99%

Regularity of Squarefree Monomial Ideals

Harbourne

2014

Springer Proceedings in Mathematics &Amp; Statistics

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“…Indeed [25] observes that the disjoint union of cyclic graphs can give arbitrarily large differences between the induced matching number, regularity, and matching number.…”

Section: Proofmentioning

confidence: 99%

“…Unfortunately, this problem is NP-hard even for vertex-decomposable graphs, as can be seen by considering a whiskered graph [25,Section 4.5]. Since computing the independence complex of a graph is already an NP-hard problem, the question of whether reg ∆ may be efficiently computed from an appropriate representation of ∆ (e.g., a list of facets) appears to still be open in general.…”

Section: Proof By Definition Of Shedding Vertex (Linkmentioning

confidence: 99%

“…In [10,Theorem 6.7], the first author and Van Tuyl showed that the regularity of the edge ideal of a graph G is at most one greater than the matching number of G. Indeed, it follows from their proof and was explicitly noticed in [25] that the regularity of the edge ideal of a graph is at most one greater than the minimum size of a maximal matching. The first goal of this note is extend this result to the edge ideal of a hypergraph, i.e., to any square-free monomial ideal.…”

Section: Introductionmentioning

confidence: 99%

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Results on the regularity of square-free monomial ideals

Harbourne

Woodroofe

2014

Advances in Applied Mathematics

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Abstract. In a 2008 paper, the first author and Van Tuyl proved that the regularity of the edge ideal of a graph G is at most one greater than the matching number of G. In this note, we provide a generalization of this result to any square-free monomial ideal. We define a 2-collage in a simple hypergraph to be a collection of edges with the property that for any edge E of the hypergraph, there exists an edge F in the 2-collage such that |E \ F | ≤ 1. The Castelnuovo-Mumford regularity of the edge ideal of a simple hypergraph is bounded above by a multiple of the minimum size of a 2-collage. We also give a recursive formula to compute the regularity of a vertex-decomposable hypergraph. Finally, we show that regularity in the graph case is bounded by a certain statistic based on maximal packings of nondegenerate star subgraphs.

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“…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].…”

Section: 32mentioning

confidence: 99%