Cohen-macaulay, Shellable and Unmixed Clutters with a Perfect Matching of König Type
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چکیده
Let C be a clutter with a perfect matching e1, . . . , eg of König type and let ∆C 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 that ∆C is pure shellable. We are able to describe in combinatorial terms when ∆C is pure. If C has no cycles of length 3 or 4, then it is shown that ∆C is pure if and only if ∆C is pure shellable (in this case ei has a free vertex for all i), and that ∆C is pure if and only if for any two edges f1, f2 of C and for any ei, one has that f1∩ei ⊂ f2∩ei or f2∩ei ⊂ f1∩ei. It is also shown that this ordering condition implies that ∆C 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 their edge ideals are facet ideals of shellable simplicial complexes. Furthermore if C is admissible and complete, then C is unmixed. We characterize the conditions in the Cohen-Macaulay criterion for bipartite graphs of Herzog and Hibi, and extend some results of Faridi about the structure of simplicial trees.
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تاریخ انتشار 2008