a b s t r a c tA graph is called very well-covered if it is unmixed without isolated vertices such that the cardinality of each minimal vertex cover is half the number of vertices. We first prove that a very well-covered graph is Cohen-Macaulay if and only if it is vertex decomposable. Next, we show that the Castelnuovo-Mumford regularity of the quotient ring of the edge ideal of a very well-covered graph is equal to the maximum number of pairwise 3-disjoint edges.
Abstract. We consider a class of graphs G such that the height of the edge ideal I(G) is half of the number V (G) of the vertices. We give Cohen-Macaulay criteria for such graphs.
We prove that for m ≥ 3, the symbolic power I (m) ∆ of a Stanley-Reisner ideal is Cohen-Macaulay if and only if the simplicial complex ∆ is a matroid. Similarly, the ordinary power I m ∆ is Cohen-Macaulay for some m ≥ 3 if and only if I ∆ is a complete intersection. These results solve several open questions on the Cohen-Macaulayness of ordinary and symbolic powers of Stanley-Reisner ideals. Moreover, they have interesting consequences on the Cohen-Macaulayness of symbolic powers of facet ideals and cover ideals.2000 Mathematics Subject Classification. Primary 13F55, Secondary 13H10.
In this paper, we prove that the arithmetical rank of a squarefree monomial ideal I is equal to the projective dimension of R/I in the following cases: (a) I is an almost complete intersection; (b) arithdeg I = reg I ; (c) arithdeg I = indeg I + 1.We also classify all almost complete intersection squarefree monomial ideals in terms of hypergraphs, and use this classification in the proof in case (c).
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