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 a binomial edge ideal of a graph G has a quadratic Gröbner basis with respect to some term order if and only if the graph G is closed with respect to a given labelling of the vertices ([10]). We also state some criteria for the closedness of a graph G that do not depend necessarily from the labelling of its vertex set.2000 Mathematics Subject Classification. Primary 13C05. Secondary 05C25.
Let K be a field and let S = K[x1, . . . , xn] be a polynomial ring over K. We analyze the extremal Betti numbers of special squarefree monomial ideals of S known as the t-spread stronglystable ideals, where t is an integer ≥ 1. A characterization of the extremal Betti numbers of such a class of ideals is given. Moreover, we determine the structure of the t-spread strongly stable idealswith the maximal number of extremal Betti numbers when t = 2.
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