Buchsbaumness of the second powers of edge ideals

Abstract

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]

1-well-covered graphs revisited

“…The graph G is locally triangle-free if G−N [v] is triangle-free for any vertex v ∈ V (G) [18]. If α(G) ≤ 2, the structure of locally triangle-free graphs belonging to W 2 is as follows.…”

“…Proposition 5.4 [18] Let G be a locally triangle-free graph in W 2 of order n. Then, (i) If α(G) = 1, then G is K n with n ≥ 2;…”

“…A characterization of triangle-dominating graphs (i.e., graphs where every triangle is also a dominating set) from W 2 in terms of forbidden configurations is presented in [18]. 1-well-covered circulant graphs are considered in [10].…”

On Gorenstein graphs

“…In this paper, we focus on Gorenstein graphs. In general, we cannot read off the Gorenstein property of a graph just from its structure since this property as usual depends on the characteristic of k (see [6,Proposition 3.1]), so we are interested in some classes of graphs such as: bipartite graphs, chordal graphs, triangle-free graphs, locally triangle-free graphs, planar graphs, and so on (see [4][5][6][7][8]13]).…”