Distributive Lattices, Bipartite Graphs and Alexander Duality

Authors:
Jürgen Herzog;Takayuki Hibi
Affiliations:
Fachbereich Mathematik und Informatik, Universität Duisburg-Essen, Essen, Germany 45117;Department of Pure and Applied Mathematics, Graduate School of Information Science and Technology, Osaka University, Toyonaka, Osaka, Japan 560-0043
Venue:
Journal of Algebraic Combinatorics: An International Journal
Year:
2005

Citing 2
Cited 8

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Dirac's theorem on chordal graphs and Alexander duality

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Cohen-Macaulay chordal graphs

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Shellable graphs and sequentially Cohen--Macaulay bipartite graphs

Journal of Combinatorial Theory Series A
Unmixed bipartite graphs and sublattices of the Boolean lattices

Journal of Algebraic Combinatorics: An International Journal
Regularity, depth and arithmetic rank of bipartite edge ideals

Journal of Algebraic Combinatorics: An International Journal
Monomial ideals and toric rings of Hibi type arising from a finite poset

European Journal of Combinatorics
Koszulness, Krull dimension, and other properties of graph-related algebras

Journal of Algebraic Combinatorics: An International Journal
Cohen-Macaulay Graphs and Face Vectors of Flag Complexes

SIAM Journal on Discrete Mathematics
Combinatorial characterizations of the Cohen-Macaulayness of the second power of edge ideals

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Abstract

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 驴 whose Stanley---Reisner ideal coincides with HP is Cohen---Macaulay, all the Cohen---Macaulay bipartite graphs will be classified.