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.
We compute one of the distinguished extremal Betti number of the binomial edge ideal of a block graph, and classify all block graphs admitting precisely one extremal Betti number.
In this note we prove that every closed graph G is up to isomorphism a proper interval graph. As a consequence we obtain that there exist linear-time algorithms for closed graph recognition.2000 Mathematics Subject Classification. Primary 13C05. Secondary 05C25.
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