1 Introduction

The study of combinatorial commutative algebra began with the pioneering work of R.P. Stanley [16] and G. Reisner [14] in 1975. The study of square-free monomial ideals grabbed the attention of researchers when it was seen in terms of simplicial complexes. Among the sub-classes of square-free monomial ideals, edge ideals of simple graphs, introduced by Villarreal in [17], stand out with their own identity.

The importance of the Cohen-Macaulay property in combinatorics comes through the proof of the “upper bound conjecture”. Since Cohen-Macaulay simplicial complexes play an essential role in the study of algebraic geometry, algebraic topology, and combinatorics, finding good classes of Cohen-Macaulay monomial ideals is always in high demand. Because, via the polarization technique, corresponding to any Cohen-Macaulay monomial ideal, we get a Cohen-Macaulay simplicial complex. Corresponding to a simplicial complex \(\Delta \), we can associate a graph G such that I(G) is Cohen-Macaulay if and only if \(\Delta \) is Cohen-Macaulay, where I(G) denotes the edge ideal of G. Therefore, the problem of classifying Cohen-Macaulay edge ideals of graphs is as difficult as classifying all Cohen-Macaulay simplicial complexes. There is evidence of edge ideals whose Cohen-Macaulay property depends on the base field. So, one can not expect a general classification, and people are interested in finding those classes of graphs whose Cohen-Macaulay property is independent of the underlying field. In this direction, some remarkable works are as follows: the classification of all Cohen-Macaulay trees by Villarreal [18]; all Cohen-Macaulay bipartite graphs by Herzog & Hibi [8]; all Cohen-Macaulay chordal graphs by Herzog, Hibi, and Zheng [9].

In the recent past, the notion of edge ideals of vertex-weighted directed graphs has been defined in [11], which is a generalization of the edge ideals of simple graphs.

A weighted oriented graph \(D_{G}\), with an underlying simple graph G, is a directed graph on the vertex set \(V(D_G):=V(G)\) with a weight function \(w:V(D_{G})\longrightarrow \mathbb {N}\), where \(\mathbb {N}\) denotes the set of positive integers. We denote the edge set of \(D_G\) by \(E(D_G)\). An edge of \(D_{G}\) is denoted by the ordered pair \((x_{i},x_{j})\), which means the direction of the edge is from \(x_{i}\) to \(x_{j}\).

Definition 1.1

Let \(D_{G}\) be a weighted oriented graph with \(V(D_{G})=\{x_{1},\ldots ,x_{n}\}\). Then the edge ideal of \(D_{G}\), denoted by \(I(D_{G})\), is defined as

$$\begin{aligned} I(D_{G}):=\big <\{x_{i}x_{j}^{w(x_j)}\mid (x_{i},x_{j})\in E(D_{G})\}\big > \end{aligned}$$

in the polynomial ring \(R=K[x_{1},\ldots ,x_{n}]\) over a field K. When \(w(x_i)=1\) for all i, \(I(D_G)\) coincides with the usual edge ideal I(G) of G. By saying \(D_{G}\) or \(I(D_{G})\) is Cohen-Macaulay, we mean the quotient ring \(R/I(D_{G})\) is Cohen-Macaulay.

The motivation behind studying weighted oriented edge ideals has its foundation in coding theory. Specifically, they appear as the initial ideals of certain vanishing ideals \(I(\mathcal {X})\) of some sets of projective points \(\mathcal {X}\) in the study of Reed-Muller-type codes (see [1, 10]). One can estimate some basic parameters and properties of Reed-Muller-type codes associated with \(\mathcal {X}\) by studying weighted oriented edge ideals. For example, if \(I(D_{G})\) is Cohen-Macaulay, then \(I(\mathcal {X})\) is Cohen-Macaulay.

Like the classification problem of Cohen-Macaulay simple graphs, people are interested in finding suitable classes of Cohen-Macaulay weighted oriented graphs (independent of the base field). For certain classes of weighted oriented edge ideals, Cohen-Macaulay ones are characterized combinatorially: (i) path and complete graphs [11]; (ii) forests [5]; (iii) bipartite graphs and graphs with perfect matching [6]; (iv) König graphs [12]; (v) cycles [15].

Chordal graphs play a significant role in the theory of edge ideals of simple graphs. This class is reflected in the celebrated Fröberg theorem [4], which states that I(G) has a linear resolution if and only if \(G^c\) (complement of G) is chordal. Again, Francisco and Van Tuyl showed that if G is a chordal graph, then R/I(G) is sequentially Cohen-Macaulay [3]. Herzog et al. gave the combinatorial characterization of Cohen-Macaulay chordal graphs in [9]. For chordal graphs, they proved that the Cohen-Macaulay property of I(G) is equivalent to the unmixed one and, thus, independent of the field K.

In this article, we characterize all Cohen-Macaulay chordal and simplicial weighted oriented graphs. In [2], the unmixed chordal and simplicial weighted oriented graphs were classified. We show in Theorem 3.1 that if the underlying graph G of a weighted oriented graph \(D_{G}\) is chordal or simplicial, then \(I(D_G)\) is Cohen-Macaulay if and only if \(I(D_G)\) is unmixed. This ensures that the Cohen-Macaulay property of edge ideals of weighted oriented chordal and simplicial graphs is independent of the base field. Our result generalizes the theorem of Herzog, Hibi, and Zheng [9] and includes a larger class of Cohen-Macaulay monomial ideals. Also, as a corollary, we identify all Gorenstein edge ideals of weighted oriented chordal and simplicial graphs.

2 Preliminaries

In this section, we recall some definitions and results related to our work. By a simple graph, we mean an undirected graph without multiple edges or loops.

Let G be a simple graph. A vertex cover C of G is a subset of V(G) such that \(e\cap C\ne \emptyset \) for all \(e\in E(G)\). A minimal vertex cover of G is a vertex cover C of G such that if \(C'\subsetneq C\), then \(C'\) can not be a vertex cover of G. Let \(D_{G}\) be a weighted oriented graph. For a vertex \(v\in V(G)\), we call \(\mathcal {N}_{G}(v):=\{u\in V(G)\mid \{u,v\}\in E(G)\}\) the neighbour set of v in G. We write \(\mathcal {N}_{G}[v]:=\mathcal {N}_{G}(v)\cup \{v\}\). A graph G is said to be complete if there is an edge between every pair of vertices. For \(A\subseteq V(G)\), we denote the induced subgraph of G on the vertex set A by G[A].

Definition 2.1

A cycle of length n, denoted by \(C_n\), is a connected graph on n vertices such that \(V(C_n)=\{x_1,\ldots ,x_n\}\) and \(E(C_n)=\{\{x_{i},x_{i+1}\}\mid 1\le i\le n\,\,\text {and}\,\, x_{n+1}=x_1\}\). A graph G is called a chordal graph if G has no induced cycle of length \(>3\).

Definition 2.2

A vertex \(v\in V(G)\) is said to be a simplicial vertex if the induced subgraph of G on \(\mathcal {N}_{G}[v]\) is complete i.e., \(G[\mathcal {N}_G[v]]\) is a complete graph and in this case, \(G[\mathcal {N}_G[v]]\) is called a simplex of G. A graph G is said to be a simplicial graph if every vertex of G is a simplicial vertex of G or is adjacent to a simplicial vertex of G.

Definition 2.3

([11]). Let \(D_{G}\) be a weighted oriented graph. Corresponding to a vertex cover C of G, consider the following sets

$$\begin{aligned}&\mathcal {L}_{1}(C)=\{x\in C \mid \exists \, (x,y)\in E(D_{G})\,\, \text {such that}\,\, y\not \in C\},\\&\mathcal {L}_{2}(C)=\{x\in C\mid x\not \in \mathcal {L}_{1}(C)\,\,\text {and}\,\, \exists \, (y,x)\in E(D_G)\,\,\text {such that}\,\, y\not \in C\},\\&\mathcal {L}_{3}(C)=C\setminus (\mathcal {L}_{1}(C)\cup \mathcal {L}_{2}(C))=\{x\in C\mid \mathcal {N}_{G}(x)\subseteq C\}. \end{aligned}$$

A vertex cover C of G is called a strong vertex cover of \(D_{G}\) if C is either a minimal vertex cover of G or for all \(x\in \mathcal {L}_{3}(C)\), there is an edge \((y, x)\in E(D_{G})\) with \(y\in \mathcal {L}_{2}(C)\cup \mathcal {L}_{3}(C)\) and \(w(y)\ne 1\).

Remark 2.4

Let \(D_G\) be a weighted oriented graph. Then the radical of \(I(D_G)\) is I(G). Thus, the minimal primes of \(I(D_G)\) are the associated primes of I(G), and they are precisely the ideals generated by the minimal vertex covers of G.

For a vertex cover C of \(D_G\), consider the following irreducible ideal associated to C

$$\begin{aligned} Q_{C}:=\big < \mathcal {L}_{1}(C)\cup \{x_{j}^{w(x_j)}\mid x_{j}\in \mathcal {L}_{2}(C)\cup \mathcal {L}_{3}(C)\}\big >. \end{aligned}$$

Theorem 2.5

([11, Theorem 25]). Let \(\mathcal {C}_{s}\) denote the set of all strong vertex covers of \(D_{G}\). Then the irredundant irreducible primary decomposition of \(I(D_{G})\) is given by

$$\begin{aligned} I(D_{G}) = \bigcap _{C\in \mathcal {C}_{s}} Q_{C}. \end{aligned}$$

Moreover, \(\{P_C\mid P_C=\big <C\big >,\,\,\text {where}\,\, C\in \mathcal {C}_{s}\}\) is the set of associated primes of \(I(D_G)\).

Theorem 2.6

([11, Theorem 31]). \(I(D_{G})\) is unmixed if and only if I(G) is unmixed and \(\mathcal {L}_{3}(C)=\emptyset \) for any strong vertex cover C of \(D_{G}\).

3 The Cohen-Macaulay classification

In this section, we will show that the Cohen-Macaulay property of weighted oriented chordal and simplicial graphs is equivalent to the unmixed property. For these classes of graphs, we also characterize the Gorenstein ones.

Theorem 3.1

Let \(D_{G}\) be a weighted oriented graph such that G is chordal or simplicial. Then \(I(D_G)\) is Cohen-Macaulay if and only if \(I(D_G)\) is unmixed.

Proof

The “only if” part of the theorem is well-known. Let \(I(D_G)\) be unmixed. Then I(G) is unmixed. Therefore, by [13, Theorem 1 and 2], every vertex of G belongs to exactly one simplex of G, and the vertex sets of simplices form a partition of V(G). Let \(H_{1},\ldots ,H_{m}\) be all the simplices of \(D_G\). Without loss of generality, let \(V(H_{1})=\{x_{1},\ldots ,x_{k_1}\}\), \(V(H_{2})=\{x_{k_1+1},\ldots , x_{k_2}\},\ldots , V(H_m)=\{x_{k_{m-1}+1},\ldots , x_{k_m}\}\) and \(x_{k_i}\) is a simplicial vertex of G belonging to the simplex \(H_{i}\) for each \(1\le i\le m\). Let \(h_{i}=x_{k_{i-1}+1}+\cdots +x_{k_i}\) for each \(1\le i\le m\), where \(k_0=0\).

Claim: The polynomials \(h_{1},\ldots ,h_{m}\) form a regular sequence on \(R/I(D_{G})\).

Proof of the claim. \(I(D_{G})\) is a monomial ideal, and so, every associated prime of \(I(D_G)\) is generated by a set of variables. Since \(I(D_G)\) is unmixed, all the associated primes of \(I(D_G)\) are minimal, and they are the associated primes of the ideal I(G), the radical of \(I(D_G)\). Thus, by Remark 2.4, the associated primes of \(I(D_G)\) are generated by the minimal vertex covers of G. Note that no minimal vertex cover of G contains the set \(V(H_{1})\) as \(\mathcal {N}_{G}[x_{k_{1}}]=V(H_{1})\). Therefore, \(h_{1}\) does not belong to any associated prime of \(I(D_G)\). Hence, \(h_1\) is a regular element on \(R/I(D_G)\). Now, we will show that \(h_{i}\) is regular on \(R/\big <I(D_G), h_1,\ldots ,h_{i-1}\big>\). Let \(I(D_G)=\bigcap _{C\in \mathcal {C}_{s}} Q_{C}\) be the primary decomposition of \(I(D_G)\) as described in Theorem 2.5. Let P be an associated prime of the ideal \(\big <I(D_{G}),h_1,\ldots ,h_{i-1}\big>\). Then \(I(D_G)\subseteq P\) implies \(Q_{C}\subseteq P\) for some \(C\in \mathcal {C}_{s}\) and hence, \(P_{C}\subseteq P\). Since \(H_{j}\) is a simplex for each \(1\le j\le m\), at least \(\vert V(H_{j})\vert -1\) elements of \(V(H_{j})\) belong to P. Therefore, for each \(1\le j\le i-1\), we have \(V(H_{j})\subseteq P\) as \(h_1,\ldots ,h_{i-1}\in P\). Now, it is clear that \(P_{C}+\big <\bigcup _{j=1}^{i-1}V(H_j)\big >\subseteq P\). Suppose \(P_{C}+\big <\bigcup _{j=1}^{i-1}V(H_j)\big >\subsetneq P\). Then P has an element g such that \(g\not \in P_{C}+\big <\bigcup _{j=1}^{i-1}V(H_j)\big>\). Then we can choose g from the polynomial ring \(K[V(G){\setminus } (C\cup V(H_1)\cup \cdots \cup V(H_{i-1}))]\). Since P is an associated prime ideal of \(\big <I(D_G),h_1,\ldots ,h_{i-1}\big>\) and \(h_1,\ldots ,h_{i-1}\) are linear forms, we can choose a homogeneous polynomial f such that \(\big <I(D_G),h_1,\ldots ,h_{i-1}\big >:f=P\) and \(g\in I(D_G):f\). Then \(I(D_G)\) will have an associated prime containing \(\big <P_C,g\big>\), which is a contradiction as \(I(D_G)\), being unmixed, has no embedded prime. Hence, any associated prime of \(\big <I(D_G),h_1,\ldots ,h_{i-1}\big>\) is of the form \(P_C+\big <\bigcup _{j=1}^{i-1}V(H_j)\big>\) for some \(C\in \mathcal {C}_{s}\). Now, from this observation, it is clear that the element \(h_{i}\) does not belong to P, otherwise \(V(H_{i})\subseteq P\), which is a contradiction as \(V(H_{i})\cap V(H_{j})=\emptyset \) for all \(1\le j\le i-1\) and C, being the minimal cover of \(D_G\), can not contain the set \(V(H_{i})\). We have chosen \(C\in \mathcal {C}_{s}\) in an arbitrary fashion and thus, \(h_{i}\) can not belong to any associated prime of \(\big <I(D_{G}),h_1,\ldots ,h_{i-1}\big>\). Hence, \(h_{i}\) is a regular element on \(R/\big <I(D_{G}),h_1,\ldots ,h_{i-1}\big>\). By induction hypothesis, we get that \(h_{1},\ldots ,h_{m}\) form a regular sequence on \(R/I(D_{G})\). Therefore, \(\textrm{depth}(R/I(D_{G}))\ge m\). Now, \(I(D_{G})\) is unmixed, which implies any associated prime of \(I(D_{G})\) is generated by some minimal vertex cover of G. Since \(V(H_{1}),\ldots , V(H_{m})\) form a partition of \(V(D_{G})\), each minimal vertex cover of G contains exactly \(\sum _{i=1}^{m}(\vert V(H_i)\vert -1)= \vert V(D_{G})\vert -m\) vertices. Thus, \(\textrm{ht}(I(D_{G}))=\vert V(D_{G})\vert -m\) and \(\textrm{dim}(R/I(D_G))=\vert V(D_{G})\vert -\textrm{ht}(I(D_G))=m\). It is well-known that \(\textrm{depth}(R/I(D_G))\le \textrm{dim}(R/I(D_G))\), which gives \(\textrm{depth}(R/I(D_G))= \textrm{dim}(R/I(D_G))=m\). Hence, \(R/I(D_G)\) is Cohen-Macaulay.

\(\square \)

Corollary 3.2

Let \(D_G\) be a weighted oriented graph such that G is chordal or simplicial. Then \(R/I(D_G)\) is Gorenstein if and only if \(D_G\) is a disjoint union of edges.

Proof

Let G be a chordal or simplicial graph such that I(G) is unmixed. Then by [13, Theorem 1 and 2], each vertex of G belongs to exactly one simplex of G, and the vertex sets of simplices form a partition of V(G). Therefore, the same argument given for chordal graphs in [9, Corollary 2.1] is also applicable to simplicial graphs. Hence, it follows from [9, Corollary 2.1] that R/I(G) is Gorenstein if and only, if G is a disjoint union of edges. Now, \(I(D_G)\) being a monomial ideal, by [7, Theorem 2.6], \(R/I(D_G)\) is Gorenstein implies R/I(G) is Gorenstein. Thus, \(D_G\) is a disjoint union of edges.

Conversely, if \(D_G\) is a disjoint union of edges, then \(I(D_G)\) is complete intersection, and hence, \(R/I(D_G)\) is Gorenstein. \(\square \)

Fig. 1
figure 1

A Cohen-Macaulay weighted oriented chordal graph \(D_G\)

Full size image

Example 3.3

Consider the weighted oriented chordal graph \(D_G\) in Fig. 1. Note that the induced subgraphs \(G[\{x_1,x_2,x_3,x_4\}]\), \(G[\{x_5,x_6,x_7\}]\), and \(G[\{x_8,x_9,x_{10}\}]\) are simplices of G and each vertex of G belongs to exactly one of these simplices. Then by [13, Theorem 2], I(G) is unmixed. Let C be a strong vertex cover of \(D_G\). If \(\{x_1,x_2,x_3,x_4\}\subseteq C\), then \(x_1\in \mathcal {L}_{3}(C)\), but, there is no vertex \(x_i\) with \(w(x_i)>1\) and \((x_i,x_1)\in E(D_G)\). This gives a contradiction to the definition of strong vertex cover. Therefore, \(\{x_1,x_2,x_3,x_4\}\not \subseteq C\) and similarly, \(\{x_5,x_6,x_7\}, \{x_8,x_9,x_{10}\}\not \subseteq C\). Hence, \(\mathcal {L}_{3}(C)=\emptyset \), and this is true for any strong vertex cover of \(D_G\). Thus, by Theorem 2.6, \(I(D_G)\) is unmixed and from Theorem 3.1, we get \(R/I(D_G)\) is Cohen-Macaulay.