A characterization of triangle-free Gorenstein graphs and Cohen–Macaulayness of second powers of edge ideals

Abstract

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

Buchsbaumness of the second powers of edge ideals

“…Conversely, if G a triangle-free graph in W 2 , then G is Gorenstein by [10,Theorem 3.4]. If G is isomorphic to Q 9 (resp.…”

“…For the second power, we proved that I(G) 2 is Cohen-Macaulay if and only if G is a triangle-free graph in W 2 (see [10]). As a consequence one can easily answer the question when I(G) 2 is generalized Cohen-Macaulay (see Theorem 1.1).…”

“…For a vertex v of G, let G v be the induced subgraph G \ ({v} ∪ N G (v)) of G. In this paper, we will call G a locally triangle-free graph if G v is triangle-free for any vertex v of G. It is worth mentioning that [16, Theorem 2.1] and [10,Theorem 4.4] suggest that G may be a locally triangle-free Gorenstein graph if I(G) 2 is Buchsbaum. So it is natural to characterize such graphs.…”

“…So it is natural to characterize such graphs. Note that they are in W 2 by [10,Proposition 3.7]. In this paper we will settle this problem when we obtain a characterization of locally triangle-free graphs in W 2 (see Theorem 3.6).…”

“…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]).…”