Licci binomial edge ideals

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Abstract

We give a complete characterization of graphs whose binomial edge ideal is licci. An important tool is a new general upper bound for the regularity of binomial edge ideals.

Introduction

Binomial edge ideals associated to simple graphs have been intensively studied in the last decade. Their algebraic and homological properties are intimately related to the combinatorics of the underlying graph. A lot of effort has been dedicated to study the Cohen-Macaulay property of these ideals. As in the case of classical edge ideals, an exhaustive classification of graphs whose binomial edge ideals are Cohen-Macaulay seems to be a hopeless task. There are successful attempts to characterize graphs with specific properties which have Cohen-Macaulay binomial edge ideals. For example, the Cohen-Macaulay property of binomial edge ideals is known for block graphs which include the trees [3] and for bipartite graphs [1]. We refer also to the papers [12], [20], [21], [22] for other classes of Cohen-Macaulay binomial edge ideals.

Let G be a simple graph (that is, undirected, with no loops, and no multiple edges) on the vertex set [n]:={1,2,,n} and S=K[x1,,xn,y1,yn] the polynomial ring in 2n variables. The binomial edge ideal JGS of G is generated by all the binomials of the form fij=xiyjxjyi where {i,j} is an edge of G. In other words, JG is generated by the 2-minors of the generic matrix X=(x1x2xny1y2yn) which correspond to the edges of G.

In this paper, we study binomial edge ideals which are in the linkage class of a complete intersection. We call such ideals licci, in brief. Besides the Cohen-Macaulay property, they satisfy some extra conditions which make possible a full characterization of graphs whose binomial edge ideals are licci. Linkage theory has a rich history in commutative algebra and algebraic geometry. Peskine and Szpiro [19] in 1974 reduced general linkage to questions on ideals over commutative algebras and after then, a lot of work has been done to develop this theory in commutative algebra and algebraic geometry. If I,J are proper ideals in a local regular ring R, they are called directly linked and we write IJ if there exists a regular sequence z=z1,,zg in IJ such that J=(z):I and I=(z):J. One says that I and J belong to the same linkage class if there exists a sequence of direct linksI=I0I1Im=J. If J is a complete intersection ideal, then I is said to be licci. The ideals in the same linkage class share several properties. For example, if I and J are linked, then I is Cohen-Macaulay if and only if J is Cohen-Macaulay. In particular, it follows that a licci ideal is Cohen-Macaulay.

The following natural question arises. May we give a full characterization of the graphs G with the property that the associated binomial edge ideal is licci?

In this paper, we give a complete answer to this question. In [9] a necessary condition for a Cohen-Macaulay homogeneous ideal in a polynomial ring to be licci is given. In the case of binomial edge ideals, this condition implies that if (JG)mSm (here m is the maximal graded ideal of the ring S) is licci, then reg(S/JG)n2. This condition turns to be also sufficient for Cohen-Macaulay binomial edge ideals as we are going to show in this paper.

The regularity of binomial edge ideals have been intensively studied in the last years. In [15] it was proved that the regularity of S/JG is upper bounded by n1 and it was conjectured that this upper bound is attained if and only if G is a path graph. This conjecture was later proved in [13]. Inspired by the paper [13], we prove a new upper bound for reg(S/JG) which is stronger than n1 and it plays an essential role in the characterization of the graphs G whose binomial edge ideal is licci.

The structure of the paper is as follows. In Section 1, we recall the basic results on licci and binomial edge ideals needed in the next sections. In Section 2, we prove that if G is a connected graph, then reg(S/JG)ndimΔ(G), where Δ(G) is the clique complex of G (Theorem 2.1). We believe that this new general upper bound for the regularity of binomial edge ideals will inspire new results on their resolution. In brief, in Theorem 2.1, we prove that for every clique W[n] of the connected graph G, we have reg(S/JG)n|W|+1. The proof is based on a double induction. First we make induction on n|W| and, secondly, on a combinatorial invariant of G.

The characterization of graphs whose binomial edge ideal is licci is given in Section 3. In Theorem 3.5 we show that, for a connected graph G on n vertices, the following statements are equivalent:

  • (i)

    (JG)mSm is licci.

  • (ii)

    JG is Cohen-Macaulay and n2reg(S/JG)n1.

  • (iii)

    G is a path graph or it is a triangle with possibly some paths attached to some of its vertices.

The most technical part in the proof is to show that there is no indecomposable graph G with n4 vertices with reg(S/JG)=n2 and JG Cohen-Macaulay. In order to make this part easier to understand, we proved some preparatory lemmas. We can reformulate the above statement by saying that the only indecomposable graphs G with JG a Cohen-Macaulay ideal and reg(S/JG)=n2 are the path with one edge and the triangle. Next we combine this fact with Lemma 3.2 which shows that for any decomposable graph G with reg(S/JG)=n2, one of the components must be a path. In this way we derive the combinatorial characterization from Theorem 3.5 (iii).

A straightforward consequence of Theorem 3.5 is Corollary 3.7 which says that for a connected bipartite graph G, the ideal (JG)mSm is licci if and only if G is a path graph. The case when G is a disconnected graph is treated in Proposition 3.8.

In the last section of the paper, we show that for chordal graphs, in the equivalent statements of Theorem 3.5, we may replace the Cohen-Macaulay property with the unmixedness of the ideal JG (Theorem 4.2). For the proof we use a theorem of Dirac which characterizes the chordal graphs in terms of their clique complex.

Section snippets

Preliminaries

We recall some notions and fundamental results needed in the later sections.

A new upper bound for the regularity of binomial edge ideals

In this section, we give a new general upper bound for the regularity of S/JG.

Theorem 2.1

Let G be a connected graph on [n]. Then reg(S/JG)ndimΔ(G).

When G is not connected, we derive the following upper bound for the regularity of S/JG.

Corollary 2.2

Let G be a graph on n vertices with the connected components G1,,Gc. Thenreg(S/JG)n(dimΔ(G1)++dimΔ(Gc)).

Let us make some short remarks before proving the above theorem. This new bound will be an essential tool in proving Theorem 3.5. Moreover, it is a substantial

Licci binomial edge ideals

As in the previous section, let G be a simple graph on the vertex set [n] and S=K[x1,,xn,y1,,yn] the polynomial ring over a field K. Let m be the maximal graded ideal of S and set R=Sm.

We recall the notion of decomposable graphs from [8].

Definition 3.1

A connected graph G is called decomposable if there exists two subgraphs G1 and G2 of G such that G=G1G2 with V(G1)V(G2)={v} where v is a simplicial vertex in G1 and G2. In this case we say that G is decomposable in the vertex v. Otherwise, the graph G is

Licci binomial edge ideals of chordal graphs

In this section we show that if we restrict to chordal graphs, we may relax the condition (ii) in Theorem 3.5, namely, we may ask that JG is only unmixed instead of being Cohen-Macaulay. Before proving the main theorem of this section, we need a preparatory result. We first recall that for a graph G, c(G) denotes the number of maximal cliques of G, that is, the number of facets of the clique complex Δ(G).

Lemma 4.1

Let G be a connected chordal graph with n vertices. Then c(G)=n2 if and only if the

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    The second author was supported by GNSAGA of INdAM (Italy). The third author was supported by the JSPS Grant-in Aid for Scientific Research (C) 18K03244.

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