Distributive Lattices, Bipartite Graphs and Alexander Duality

By Jürgen Herzog, Takayuki Hibi. 2003

214 citations recorded by the source corpus.

A certain squarefree monomial ideal HP arising from a finite partially ordered set P will be studied from viewpoints of both commutative algbera and combinatorics. First, it is proved that the defining ideal of the Rees algebra of HP possesses a quadratic Gröbner basis. Thus in particular all powers of HP have linear resolutions. Second, the minimal free graded resolution of HP will be constructed explicitly and a combinatorial formula to compute the Betti numbers of HP will be presented. Third, by using the fact that the Alexander dual of the simplicial complex Δ whose Stanley–Reisner ideal coincides with HP is Cohen–Macaulay, all the Cohen–Macaulay bipartite graphs will be classified.

Subfields: Algebra and Number Theory, Computational Theory and Mathematics, Discrete Mathematics and Combinatorics

View Paper: https://www.semanticscholar.org/paper/dad4a5e0d45758b936967c34311e757761575005

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Citations in the corpus

Construction of Cohen–Macaulay Binomial Edge Ideals (2012, methodology): “Many authors have focused their attention on such ideals (see for example [5, 10]).”
The depth of powers of an ideal (2004, background): “In the previous paper [11] it was proved that each powerH P has a linear resolution.”
Regularity, depth and arithmetic rank of bipartite edge ideals (2009, background): “(Note that I is the ideal HdG , in the notation of [4], with the xi and the yj interchanged.”
Cohen-Macaulay edge ideal whose height is half of the number of vertices (2009, methodology): “We also have a Cohen-Macaulay criterion which is similar to Herzog-Hibi ([8], Theorem 3.”
Sequentially Cohen–Macaulay bipartite graphs: vertex decomposability and regularity (2009, background): “VG as VG = V1 ∪ V2 so that every edge e ∈ EG has one endpoint in V1 and the other endpoint in V2. SCM bipartite graphs, which includes the set of Cohen-Macaulay bipartite graphs, have been studied in [1, 5, 6, 7, 9, 12, 13, 20]. With the additional assumption that G is bipartite, these papers have s”
Vertex decomposability and regularity of very well-covered graphs (2010, background): “Unmixed bipartite graphs and Cohen-Macaulay bipartite graphs have been characterized nicely in terms of combinatorial properties of G (see [7] and [14]).”
Edge ideals: algebraic and combinatorial properties (2010, background): “e Cohen-Macaulay and unmixed properties. There are classiﬁcations of the following families in terms of combinatorial properties of the graph or clutter: (c1) [94, 111] unmixed bipartite graphs, (c2) [36, 60] Cohen-Macaulay bipartite graphs, (c3) [109] Cohen-Macaulay trees, (c4) [38] totally balance”