COHEN-MACAULAY, SHELLABLE AND UNMIXED CLUTTERS WITH A PERFECT MATCHING OF K ¨

By Susan Morey, Enrique Reyes, Rafael H. Villarreal. 2007

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Let C be a clutter with a perfect matching e1, . . . , eg of Konig type and letC be the Stanley-Reisner complex of the edge ideal of C. If all c-minors of C have a free vertex and C is unmixed, we show thatC is pure shellable. We are able to describe, in combinatorial and algebraic terms, whenC is pure. If C has no cycles of length 3 or 4, then it is shown thatC is pure if and only ifC is pure shellable (in this case ei has a free vertex for all i), and thatC is pure if and only if for any two edges f1, f2 of C and for any ei, one has that f1\eif2\ei or f2\eif1\ei. It is also shown that this ordering condition implies thatC is pure shellable, without any assumption on the cycles of C. Then we prove that complete admissible uniform clutters and their Alexander duals are unmixed. In addition, the edge ideals of complete admissible uniform clutters are facet ideals of shellable simplicial complexes, they are Cohen- Macaulay, and they have linear resolutions. Furthermore if C is admissible and complete, then C is unmixed. We characterize certain conditions that occur in a Cohen-Macaulay criterion for bipartite graphs of Herzog and Hibi, and extend some results of Faridi—on the structure of unmixed simplicial trees—to clutters with the Konig property without 3-cycles or 4-cycles.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/f20cf1bbf772ca0d6f60d1d52127dda95e427cf4

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Citations in the corpus

Edge ideals: algebraic and combinatorial properties (2010, background): “1} with Fj \ Fℓ = {v}. We are interested in determining which families of clutters have the property that ∆C is pure, Cohen-Macaulay, or shellable. These properties have been extensively studied, see [10, 89, 92, 93, 94, 100, 102, 110, 111] and the references there. The above deﬁnition of shellable ”
Cohen-Macaulay edge ideal whose height is half of the number of vertices (2009, background): “−yn} is a system of parameters of S/I(G). For later use we give a characterization of the unmixedness for our graphs, that is a more detailed description, but a special case of a more general result ([10], Theorem2.9): Proposition 2.3. Let G be a graph with 2n vertices, which are not isolated, and w”