Linearity defect of edge ideals and Fröberg’s theorem

Abstract

Abstract. Fröberg's classical theorem about edge ideals with 2-linear resolution can be regarded as a classification of graphs whose edge ideals have linearity defect zero. Extending his theorem, we classify all graphs whose edge ideals have linearity defect at most 1. Our characterization is independent of the characteristic of the base field: the graphs in question are exactly weakly chordal graphs with induced matching number at most 2. The proof uses the theory of Betti splittings of monomial ideals due to Francisco, Hà, and Van Tuyl and the structure of weakly chordal graphs. Along the way, we compute the linearity defect of edge ideals of cycles and weakly chordal graphs. We are also able to recover and generalize previous results due to Dochtermann-Engström, Kimura and Woodroofe on the projective dimension and Castelnuovo-Mumford regularity of edge ideals. IntroductionLet (R, m) be a standard graded algebra over a field k with the graded maximal ideal m. Let M be a finitely generated graded R-module. For integers i, j,The regularity is an important invariant of graded modules over R. When R is a polynomial ring and M is a monomial ideal of R, its regularity exposes many combinatorial flavors. This fact has been exploited and proved to be very useful for studying the regularity; for recent surveys, see [13], [28], [38]. A classical and instructive example is Fröberg's theorem. Recall that if G is a graph on the vertex set {x 1 , . . . , x n } (where n ≥ 1), and by abuse of notation, R is the polynomial ring k[x 1 , . . . , x n ], then the edge ideal of G is I(G) = (x i x j : {x i , x j } is an edge of G). Unless otherwise stated, whenever we talk about an invariant of I(G) (including the regularity and the linearity defect, to be defined below), it is understood that the base ring is the polynomial ring R. For m ≥ 3, the cycle C m is the graph on vertices x 1 , . . . , x m with edges x 1 x 2 , . . . , x m−1 x m , x m x 1 . We say that a graph G is weakly chordal (or weakly triangulated) if for every m ≥ 5, neither G, nor its complement contains C m as an induced subgraph. G is chordal if for any m ≥ 4, C m is not an induced subgraph of G. Fröberg's theorem [10] says that I(G) has regularity 2 if and only if the complement graph of G is chordal. It is of interest to find generalizations of this important result; see, for instance, [7], [8]. Theorem 4.1] extended Fröberg's theorem by providing a combinatorial characterization of connected bipartite graphs G such that reg I(G) = 3. In general, the regularity 3 condition on edge ideals is not 2010 Mathematics Subject Classification. 05C25, 13D02, 13H99.

Powers of sums and their homological invariants

“…Morphisms of both type are well-suited to study the linearity defect but the latter yields more precise information. We also recall some results from [40], [42], which will be used frequently later.…”

“…The following are equivalent: (i) The decomposition P = I + J is a Betti splitting; (ii) The morphisms I ∩ J → I and I ∩ J → J are Tor-vanishing; (iii) The mapping cone construction for the map I ∩ J → I ⊕ J yields a minimal free resolution of P .Most results of this paper are motivated by the next simple observation, presented in[42, Example 4.7].…”

“…[42, Theorem 4.9]). Let P = I +J be a Betti splitting of non-zero proper ideals of R. Then there are inequalities…”

Nonvanishing Betti Numbers of Weakly Chordal Graphs

“…Let imn(G) denote the maximum number of parwise 3-disjoint edges of G. Further, following [13], we define d(G) = max{ r i=1 |V (B i )| − r}, where the maximum is taken over all the strongly disjoint families {B 1 , . .…”

Induced matchings in strongly biconvex graphs and some algebraic applications

“…Theorem Let G be a weakly chordal graph on n vertices. Then the following statements hold: (a) ([19, Theorem 1.1], [20, Theorem 3.4]) if and only if there exists with and . (b) [25, Theorem 14] . (c) [21, Theorem 7.7] . …”