Monomial ideals of weighted oriented graphs

Let I=I(D) be the edge ideal of a weighted oriented graph D. We determine the irredundant irreducible decomposition of I. Also, we characterize the associated primes and the unmixed property of I. Furthermore, we give a combinatorial characterization for the unmixed property of I, when D is bipartite, D is a whisker or D is a cycle. Finally, we study the Cohen-Macaulay property of I.


Introduction
, respectively. Some times for short we denote these sets by V and E respectively. The weight of x ∈ V is w(x). If e = (x, y) ∈ E, then x is the tail of e and y is the head of e. The underlying graph of D is the simple graph G whose vertex set is V and whose edge set is {{x, y}|(x, y) ∈ E}. If V (D) = {x 1 , . . . , x n }, then we consider the polynomial ring R = K[x 1 , . . . , x n ] in n variables over a field K. In this paper we introduce and study the edge ideal of D given by I(D) = (x i x w(xj ) j : (x i , x j ) ∈ E(D)) in R, (see Definition 3.1).
In Section 2 we study the vertex covers of D. In particular we introduce the notion of strong vertex cover (Definition 2.6) and we prove that a minimal vertex cover is strong. In Section 3 we characterize the irredundant irreducible decomposition of I(D). In particular we show that the minimal monomial irreducible ideals of I(D) are associated with the strong vertex covers of D. In Section 4 we give the following characterization of the unmixed property of I(D).

I(D) is unmixed
G is unmixed and D has the minimal-strong property Furthermore, if D is bipartite, D is a whisker or D is a cycle, we give an effective (combinatorial) characterization of the unmixed property. Finally in Section 5 we study the Cohen-Macaulayness of I(D). In particular we characterize the Cohen-Macaulayness when D is a path or D is complete. Also, we give an example where this property depend of the characteristic of the field K.

Weighted oriented graphs and their vertex covers
In this section we define the weighted oriented graphs and we study their vertex covers. Furthermore, we define the strong vertex covers and we characterize when V (D) is a strong vertex cover of D. In this paper we denote the set {x ∈ V | w(x) = 1} by V + . Definition 2.1. A vertex cover C of D is a subset of V , such that if (x, y) ∈ E, then x ∈ C or y ∈ C. A vertex cover C of D is minimal if each proper subset of C is not a vertex cover of D. Proof. If x ∈ L 3 (C), then N + D (x) ⊆ C, since x / ∈ L 1 (C). Furthermore N − D (x) ⊆ C, since x / ∈ L 2 (C). Hence N D (x) ⊂ C, since x / ∈ N D (x). Now, if x ∈ C and N D (x) ⊂ C, then x / ∈ L 1 (C) ∪ L 2 (C). Therefore x ∈ L 3 (C).
Thus, there is y ∈ N D (x) \ C implying C \ {x} is not a vertex cover. Therefore, C is a minimal vertex cover.
Remark 2.7. Let C be a vertex cover of D. Hence, by Proposition 2.4 and since Corollary 2.8. If C is a minimal vertex cover of D, then C is strong.
Definition 2.11. D is called unicycle oriented graph if it satisfies the following conditions: 1) The underlying graph of D is connected and it has exactly one cycle C.
2) C is an oriented cycle in D. Furthermore for each y ∈ V (D) \ V (C), there is an oriented path from C to y in D.
Proof. We take x ∈ V (D ′ ). Thus, by Remark 2.9, there is y ∈ N − D (x) ∩ V + (D). If y ∈ D 1 , then we take D 2 = D 1 ∪ {(y, x)}. Hence, if C is the oriented cycle of D 1 , then C is the unique cycle of D 2 , since deg D2 (x) = 1. If y ∈ C, then (y, x) is an oriented path from C to x. Now, if y / ∈ C, then there is an oriented path L form C to y in D 1 . Consequently, L ∪ {(y, x)} is an oriented path form C to x. Furthermore, deg D2 (x) = 1 and w(y) = 1, then D 2 is an unicycle oriented graph. A contradiction, since D 1 is maximal. This implies y ∈ V (D ′ ), so y ∈ N − D ′ (x) ∩ V + (D ′ ). Therefore, by Remark 2.9, V (D ′ ) is a strong vertex cover of D ′ . Lemma 2.13. If V (D) is a strong vertex cover of D, then there is an unicycle oriented subgraph of D.

⇐)
We take x ∈ V (D). By hypothesis there is 1 ≤ j ≤ s such that x ∈ V (D j ). We assume C is the oriented cycle of D j . If x ∈ V (C), then there is y ∈ V (C) such that (y, x) ∈ E(D j ) and w(y) = 1, since deg Dj (y) ≥ 2 and D j is a unicycle oriented subgraph. Now, we assume x / ∈ V (C), then there is an oriented path L = (z 1 , . . . , z r ) such that z 1 ∈ V (C) and z r = x. Thus, (z r−1 , x) ∈ E(D). Furthermore, w(z r−1 ) = 1, since deg Dj (z r−1 ) ≥ 2. Therefore V is a strong vertex cover.

Edge ideals and their primary decomposition
As is usual if I is a monomial ideal of a polynomial ring R, we denote by G(I) the minimal monomial set of generators of I. Furthermore, there exists a unique decomposition, I = q 1 ∩ · · · ∩ q r , where q 1 , . . . , q r are irreducible monomial ideals such that I = i =j q i for each j = 1, . . . , r. This is called the irredundant irreducible decomposition of I. Furthermore, q i is an irreducible monomial ideal if and only if q i = (x a1 i1 , . . . , x as is ) for some variables x ij . Irreducible ideals are primary, then a irreducible decomposition is a primary decomposition. For more details of primary decomposition of monomial ideals see [6,Chapter 6]. In this section, we define the edge ideal I(D) of a weighted oriented graph D and we characterize its irredundant irreducible decomposition. In particular we prove that this decomposition is an irreducible primary decomposition, i.e, the radicals of the elements of the irredundant irreducible decomposition of I(D) are different.
. A sink of D is a vertex y such that N D (y) = N − D (y). Remark 3.3. Let D = (V, E, w) be a weighted oriented graph. We take D ′ = (V, E, w ′ ) a weighted oriented graph such that w ′ (x) = w(x) if x is not a source and w ′ (x) = 1 if x is a source. Hence, I(D) = I(D ′ ). For this reason in this paper, we will always assume that if x is a source, then w(x i ) = 1.
Definition 3.4. Let C be a vertex cover of D, the irreducible ideal associated to C is the ideal Proof. We take I = I(D) and m ∈ G(I), then m = xy w(y) , where (x, y) ∈ D. Since C is a vertex cover, x ∈ C or y ∈ C. If y ∈ C, then y ∈ I C or y w(y) ∈ I C . Thus, m = xy w(y) ∈ I C . Now, we assume y / ∈ C, then x ∈ C. Hence, y ∈ N + D (x) ∩ C c , so x ∈ L 1 (C). Consequently, x ∈ I C implying m = xy w(y) ∈ I C . Therefore I ⊆ I C .
Definition 3.6. Let I be a monomial ideal. An irreducible monomial ideal q that contains I is called a minimal irreducible monomial ideal of I if for any irreducible monomial ideal p such that I ⊆ p ⊆ q one has that p = q. 2) There is a strong vertex cover C of D such that J = I C .
Hence, I(D) ⊆ J ′ . This is a contradiction, since J is minimal. Therefore C is strong.
A contradiction, since x αu ju / ∈ I C . Therefore I(D) = C∈Cs I C is the irredundant irreducible decomposition of I(D).
Remark 3.12. If C 1 , . . . , C s are the strong vertex covers of D, then by Theorem 3.11, I C1 ∩ · · · ∩ I Cs is the irredundant irreducible decomposition of I(D). Furthermore, if P i = rad(I Ci ), then P i = (C i ). So, P i = P j for 1 ≤ i < j ≤ s. Thus, I C1 ∩ · · · ∩ I Cs is an irredundant primary decomposition of I(D). In particular we have Ass(I(D)) = {P 1 , . . . , P s }. Example 3.13. Let D be the following oriented weighted graph x 3 x 4 x 5 x 1 . From Theorem 3.10 and Theorem 3.11, the irreducible decomposition of I(D) is: . Example 3.14. Let D be the following oriented weighted graph Hence, I(D) = (x 1 x 2 2 , x 2 x 5 3 , x 3 x 7 4 ). By Theorem 3.10 and Theorem 3.11, the irreducible decomposition of I(D) is: In Example 3.13 and Example 3.14, I(D) has embedding primes. Furthermore the monomial ideal (V (D)) is an associated prime of I(D) in Example 3.13. Proposition 2.14 and Remark 3.12 give a combinatorial criterion for to decide when (V (D)) ∈ Ass(I(D)).

Unmixed weighted oriented graphs
Let D = (V, E, w) be a weighted oriented graph whose underlying graph is G = (V, E). In this section we characterize the unmixed property of I(D) and we prove that this property is closed under c-minors. In particular if G is a bipartite graph or G is a whisker or G is a cycle, we give an effective (combinatorial) characterization of this property.  2) Each strong vertex cover of D has the same cardinality.
3) I(G) is unmixed and L 3 (C) = ∅ for each strong vertex cover C of D.
. Also, w(x j ) = 1 and x j ∈ V = L 3 (V ). So, V is a strong vertex cover. Hence, by Proposition 4.9, I(D) is mixed.
In the following three results we assume that D 1 , . . . , D r are the connected components of D. Furthermore G i is the underlying graph of D i .
, since x ∈ D j . Consequently, by Lemma 4.11, z ∈ C j \ L 1 (C j ). Therefore C j is strong.

⇐)
We take x ∈ L 3 (C), then x ∈ C i for some 1 ≤ i ≤ r. Then, by Lemma 4.11, x ∈ L 3 (C i ). Thus, there is a ∈ N − Di (x) such that w(a) = 1 and a ∈ C i \ L 1 (C i ), since C i is strong. Hence, by Lemma 4.11, a ∈ C \ L 1 (C). Therefore C is strong.    2) If (x i , y i ) ∈ E(H) for some 1 ≤ i ≤ n, then w(x i ) = 1.
Proof. 2) ⇒ 1) We take a strong vertex cover C of H. Suppose x j , y j ∈ C, then y j ∈ L 3 (C), since N D (y j ) = {x j } ⊆ C. Consequently, (x j , y j ) ∈ E(G) and w(x j ) = 1, since C is strong. This is a contradiction by condition 2). This implies, |C ∩ {x i , y i }| = 1 for each 1 ≤ i ≤ n. So, |C| = n. Therefore, by Theorem 4.2, I(H) is unmixed. 1) ⇒ 2) By contradiction suppose (x i , y i ) ∈ E(H) and w(x i ) = 1 for some i. Since w(x i ) = 1 and by Remark 3.3, we have that x i is not a source. Thus, there is x j ∈ V (D), such that (x j , x i ) ∈ E(H). We take the vertex cover Proof. ⇐) By 1) and [1, Theorem 2.5.7], G is unmixed. We take a strong vertex cover and C is strong. Hence L 3 (C) = ∅ and D has the strong-minimal property. Therefore I(D) is unmixed, by Theorem 4.2.
If w(x k j ) = 1, then we take Proof. We take a strong vertex cover C of D. Hence, if y ∈ L 3 (C), then there is (z, y) ∈ E(D) with z ∈ V + . Consequently, by hypothesis, z is a sink. A contradiction, since (z, y) ∈ E(D). Therefore, L 3 (C) = ∅ and C is a minimal vertex cover. ⇒) By contradiction, suppose there is (z, y) ∈ E(D), with z ∈ V + . We can assume G = (x 1 , x 2 , . . . , x n , x 1 ) ≃ C n , with x 2 = y and x 3 = z. We take a strong vertex cover C in the following form: Then, x 3 ∈ L 2 (C). This implies C is a strong vertex cover. But L 3 (C) = ∅. A contradiction, since D has the minimal-strong property.

Cohen-Macaulay weighted oriented graphs
In this section we study the Cohen-Macaulayness of I(D). In particular we give a combinatorial characterization of this property when D is a path or D is complete. Furthermore, we show the Cohen-Macaulay property depends of the characteristic of K.  Example 5.4. In Example 3.13 and Example 3.14 I(D) is mixed. Hence, I(D) is not Cohen-Macaulay, but I(G) is Cohen-Macaulay.

Conjecture 5.5. I(D) is Cohen-Macaulay if and only if I(G) is Cohen-Macaulay and D has the minimal-strong property. Equivalently I(D) is Cohen-Macaulay if and only if I(D) is unmixed and I(G) is Cohen-Macaulay.
Proposition 5.6. Let D be a weighted oriented graph such that V = {x 1 , . . . , x k } and whose underlying graph is a path G = (x 1 , . . . , x k ). Then the following conditions are equivalent.
Theorem 5.7. If G is a complete graph, then the following conditions are equivalent.