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Two-dimensional squarefree monomial ideals can be seen as the Stanley-Reisner ideals of graphs. The main results of this paper are combinatorial characterizations for the Cohen-Macaulayness of ordinary and symbolic powers of such an ideal in terms of the associated graph.

We classify Cohen–Macaulay graphs of girth at least 5 and planar Gorenstein graphs of girth at least 4. Moreover, such graphs are also vertex decomposable.

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a b s t r a c tLet S be a polynomial ring and I be the Stanley-Reisner ideal of a simplicial complex ∆. The purpose of this paper is investigating the Buchsbaum property of S/I (r) when ∆ is pure dimension 1. We shall characterize the Buchsbaumness of S/I (r) in terms of the graphical property of ∆. That is closely related to the characterization of the Cohen-Macaulayness of S/I (r) due to the first author and N.V. Trung. Crown