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.

Abstract. We graph-theoretically characterize triangle-free Gorenstein graphs G.As an application, we classify when I(G) 2 is Cohen-Macaulay.

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We study the equality of the extremal Betti numbers of the binomial edge ideal and those of its initial ideal in( ) for a closed graph . We prove that in some cases there is a unique extremal Betti number for in( ) and as a consequence there is a unique extremal Betti number for and these extremal Betti numbers are equal. K E Y W O R D SClosed graphs, binomial edge ideals, projective dimension, Betti numbers M S C ( 2 0 1 0 ) 05E40, 13C14, 13C15

Abstract. We graph-theoretically characterize the class of graphs G such that I(G) 2 are Buchsbaum. IntroductionThroughout this paper let G = (V (G), E(G)) be a finite simple graph without isolated vertices. An independent set in G is a set of vertices no two of which are adjacent to each other. The size of the largest independent set, denoted by α(G), is called the independence number of G. A graph is called well-covered if every maximal independent set has the same size. A well-covered graph G is a member of the class W 2 if the remove any vertex of G leaves a well-covered graph with the same independence number as G (see e.g. [14]).Let R = K[x 1 , . . . , x n ] be a polynomial ring of n variables over a given field K. Let G be a simple graph on the vertex set V (G) = {x 1 , . . . , x n }. We associate to the graph G a quadratic squarefree monomial idealwhich is called the edge ideal of G. We say that G is Cohen-Macaulay (resp. Gorenstein) if I(G) is a Cohen-Macaulay (resp. Gorenstein) ideal. It is known that G is well-covered whenever it is Cohen-Macaulay (see e.g. [20, Proposition 6.1.21]) and G is in W 2 whenever it is Gorenstein (see e.g. [10, Lemma 2.5]). It is a wide open problem to characterize graph-theoretically the Cohen-Macaulay (resp. Gorenstein) graphs. This problem was considered for certain classes of graphs (see [5,6,9,10]). Generally, we cannot read off the Cohen-Macaulay and Gorenstein properties of G just from its structure because these properties in fact depend on the characteristic of the base field K (see [20, Exercise 5.3.31] and [10, Proposition 2.1]).If we move on to the higher powers of I(G), then we can graph-theoretically characterize G such that I(G) m is Cohen-Macaulay (or Buchsbaum, or generalized CohenMacaulay) for some m 3 (and for all m 1) (see [4,15,19]

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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