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Koszulness of Vertex Cover Algebras of Bipartite GraphsGiancarlo Rinaldo aa Dipartimento di Matematica , Universitá di Messina , Messina , ItalyPublished online: 20 Jul 2011.

To cite this article: Giancarlo Rinaldo (2011) Koszulness of Vertex Cover Algebras of Bipartite Graphs, Communications inAlgebra, 39:7, 2249-2259, DOI: 10.1080/00927870903286884

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Communications in Algebra®, 39: 2249–2259, 2011Copyright © Taylor & Francis Group, LLCISSN: 0092-7872 print/1532-4125 onlineDOI: 10.1080/00927870903286884

KOSZULNESS OF VERTEX COVERALGEBRAS OF BIPARTITE GRAPHS

Giancarlo RinaldoDipartimento di Matematica, Universitá di Messina, Messina, Italy

Let G be a bipartite graph and let �A�G� be the fiber cone of the Alexander dual ofthe edge ideal. We show that when �A�G� is a domain then �A�G� is Koszul. We alsogive new characterizations of unmixed and Cohen–Macaulay bipartite graphs.

Key Words: Cohen–Macaulay; Edge ideal; Fiber cone; Koszul; Vertex cover.

2000 Mathematics Subject Classification: Primary 13F55; Secondary 13H10.

INTRODUCTION

Let G be a graph with vertex set V�G� = �1� � � � � n� without isolated verticesand edge set E�G�. Consider the polynomial ring S = K�x1� � � � � xn� over a field K,� = �x1� � � � � xn� the maximal irrelevant ideal of S. The edge ideal of the graph G is

I = �xixj � �i� j� ∈ E�G���

A second monomial ideal related to G is the Alexander dual of I that is

J = ⋂�i�j�∈E�G�

�xi� xj��

whose minimal set of monomial generators describes exactly the minimal vertexcovers of G.

A k-cover of G is a vector a = �a�1�� � � � � a�n�� ∈ �n, with k ∈ �, that satisfiesthe condition a�i�+ a�j� ≥ k for all �i� j� ∈ E�G�. The vertex cover algebra A�G� isthe subalgebra of S�t� generated by all monomials of the form xatk, where a is ak-cover of G.

In [7], Herzog, Hibi, and Trung showed that A�G� is standard graded if andonly if G is a bipartite graph (see also [4] for a result complementing this one). Thisproperty is induced by the decomposability of all k-covers with k ≥ 2. In this case,

Received March 24, 2009; Revised July 18, 2009. Communicated by R. Villarreal.Address correspondence to Dr. Giancarlo Rinaldo, Dipartimento di Matematica, Universitá di

Messina, Salita Sperone, 31. S. Agata, Messina 98166, Italy; E-mail: grinaldo@unime.it

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the algebra A�G� is the Rees algebra of the ideal J . A second classical algebra thatwe can define on J is its fiber cone which, under our assumption is

�A�G� = A�G�/�A�G��

Benedetti, Costantinescu, and Varbaro [1] characterized when �A�G� is a domainfrom the combinatorics of the graph G. The principal aim of our work is to provethat when �A�G� is a domain, then it is Koszul. To reach this goal, we observe thatif �A�G� is a domain, then it is isomorphic to �A�G′�, where G′ is a Cohen–Macaulaybipartite graph.

In Section 1, we recall some basic concepts and define notations on graphsand k-covers. In Section 2, we focus our attention on an interesting subgraph ofG: the graph whose edges are the set of nice matchings. We show that a connectedset of nice matchings is a complete bipartite graph (see Proposition 2.4). By thisobservation, we give new characterizations of unmixed (Theorem 2.8) and Cohen–Macaulay (Theorem 2.10) bipartite graphs. We recall that the first characterizationof unmixed bipartite graphs was given by Ravindra (see [8]) and the second one byVillarreal (see [11]). The first characterization of Cohen–Macaulay bipartite graphswas given by Herzog and Hibi (see [5]). By Theorems 2.8 and 2.10, and observingthat “overlapping” adjacent edges that are nice matchings of G, we obtain a newbipartite graph G′ such that �A�G� � �A�G′� (Lemma 3.6). In Section 3, we obtainthe main results (Theorem 3.7 and Corollary 3.9).

1. PRELIMINARIES

In this section, we recall some concepts and notations on graphs and k-coversthat we shall use in the article.

Let G be a graph on the vertex set V�G� = �n� = �1� � � � � n� without loops andhaving no multiple edges and E�G� its set of edges. A graph G is bipartite if itsvertex set V�G� can be partitioned into two subsets V1 and V2 such that every edgein E�G� joins a vertex in V1 with a vertex in V2. If every vertex of V1 is adjacent toevery vertex of V2 the graph is called a complete bipartite graph: if V1 and V2 have mand n vertices, respectively, we write G = Km�n. A subgraph of G is a graph havingall its vertices and edges in G. The induced subgraph G�W by W ⊂ V�G� is defined by

G�W = �W� �e ∈ E�G� e ⊂ W���

A subgraph H of G spans G if V�H� = V�G�.The neighborhood of u ∈ V�G� is the set

NG�u� = �v ∈ V�G� �u� v� ∈ E�G���

The closed neighborhood NG�u� = NG�u� ∪ �u�. The neighbors of A ⊂ V�G�, denotedby NG�A�, are the neighbors in V�G�\A of vertices in A. The closed neighbors of Ais NG�A� = N�A� ∪ A. A subset C ⊂ V�G� is a vertex cover of G if every edge of Gis incident with a vertex in C. A vertex cover C of G is called minimal if there isno proper subset of C, which is a vertex cover of G. Let ��G� = �C1� � � � � Cr� bethe set of minimal covers of G. A graph is called unmixed if all its minimal vertex

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covers of have the same cardinality. A subset of vertices of V�G� is independent ifno two of them are adjacent. An independent set is maximal if it is not a subsetof any other independent set. A graph is said to be well covered if all its maximalindependent sets are of the same cardinality. A matching M ⊂ E�G� in G is a set ofpairwise non-adjacent edges (no two edges share a common vertex). A matching isperfect if every vertex of the graph is incident to exactly one edge of the matching.

An integer vector a = �a�1�� � � � � a�n�� is a k-cover if and only if:

(1) a�i� ≥ 0, k ≥ 0;(2) There exists i such that a�i� > 0;(3) For all �i� j� ∈ E�G�, a�i�+ a�j� ≥ k.

Hence a vertex cover corresponds to a 1-cover. A k-vertex cover a is decomposable if

a = b+ c�

where b is an h-cover and c is a �k− h�-cover. A k-cover is nonbasic if a= b+ c,where b is a k-cover and c is a 0-cover. Let ��G� = �g1� � � � � gr� be the set ofbasic 1-covers of G. To each Ci ∈ ��G� corresponds a basic 1-cover gi ∈ ��G� fori= 1� � � � � r. That is,

gi�j� ={1 if j ∈ Ci

0 if j Ci�

Let S = K�x1� � � � � xn� be a polynomial ring over a field K. The edge ideal ofG is the ideal

I = �xixj �i� j� ∈ E�G���

A graph G is called Cohen–Macaulay if S/I�G� is a Cohen–Macaulay ring. See [10]for detailed information on this subject.

2. K -VERTEX COVERS, NICE MATCHINGS AND EDGE IDEALS

We use the same notation as in the previous section unless otherwise specified.It is known (see [10], Chapter 6) that there exists a bijection between the minimalcovers of G and minimal prime ideals associated to I , that is,

I =r⋂

i=1

�xj j ∈ Ci��

Definition 2.1. We say �i� j� ∈ E�G� is a nice matching if ∀ basic 1-cover g ∈��G�we have that g�i�+ g�j� = 1. Let MG the subgraph of G with edge set

E�MG� = ��i� j� is a nice matching��

We call MG the graph of nice matchings of G.

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Example 2.2. The hexagon has no nice matchings. If G is a complete bipartite,graph, then E�MG� = E�G� (see Proposition 2.4). If G is a bipartite Cohen–Macaulay graph then E�MG� is the unique set of perfect matchings (seeTheorem 2.10).

Remark 2.3. Let G be a graph that is a disjoint union of complete bipartite graphs

G =s⋃

i=1

Kmi�ni�

Then for each C ∈ ��G�

C =s⋃

i=1

Ci�

where each Ci is one of the two minimal coverings of Kmi�ni, hence �Ci� ∈ �mi� ni�.

Proposition 2.4. Each connected component of MG is a complete bipartite graph.

Proof. Let H be a connected component of MG, and let g = �g�1�� � � � g�n�� ∈��G�. We define

C1 = �k ∈ V�H� g�k� = 1�� C0 = �k ∈ V�H� g�k� = 0��

We observe that C1 is an independent set of vertices in H . In fact, an edgeconnecting two vertices of C1 is not a nice matching.

We also observe that C0 is an independent set of vertices of G since g is a 1-cover of G. Hence C0 is an independent set of vertices of H . Therefore, H is bipartiteon the vertex set C0 ∪ C1. We claim that all minimal covers of G contain either C0

or C1 (mutually exclusive).In fact, if there exists g′ ∈ ��G� such that g′�i� = g′�j� = 1 with i ∈ C0, j ∈ C1,

we have an odd path of nice matchings from i to j

i = i0� i1� � � � � i2k−1 = j�

Hence

g′�i0� = 1� g′�i1� = 0� g′�i2� = 1� � � � � g′�i2k−1� = 0�

But this is a contradiction.Finally, we prove that if i ∈ C0, j ∈ C1, then �i� j� ∈ E�MG�. In fact, if we

consider the primary decomposition of the edge ideal, I�G� = ⋂C∈��G� PC , either xi

or xj is in PC (mutually exclusive), that is, xixj ∈ I�G� and �i� j� is a nice matching.�

Example 2.5. In the picture, we show an example (see Proposition 2.4) of a graphG and its subgraph of nice matchings MG.

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Lemma 2.6. If MG spans G, then all the basic 1-covers of G are basic 1-coversof MG.

Proof. If g = �g�1�� � � � � g�n�� ∈ ��G�, then it is a 1-cover for MG since it is asubgraph of G.

Suppose that �g�1�� � � � � g�i− 1�� g�i�− 1� g�i+ 1�� � � � � g�n�� is a 1-cover forMG. Since MG spans G there exists an edge �i� j� ∈ E�MG�. Hence g�j� > 0.Therefore, g�i�+ g�j� = 2. This is a contradiction. �

Lemma 2.7. Let G be a bipartite graph. For all �i� j� ∈ E�MG� and for all basic k-covers a, we have

a�i�+ a�j� = k�

Proof. The case k = 1 follows by the definition. If k > 1 (see Theorem 5.1.(b), [7]),we observe that each basic k-cover of a bipartite graph is a sum of basic 1-coversand the assertion follows. �

We have two nice characterizations of unmixed bipartite graphs.The first one is due to Ravindra (see [8]): G is well covered if and only if for

every edge �i� j� in the perfect matching the graph induced by the vertices NG�x� ∪NG�y� is a complete bipartite graph.

The second one is due to Villarreal (see [11]): G is unmixed if and only if thereexists a bipartition V1 = �x1� � � � � xg�, V2 = �y1� � � � � yg� of G such that (a) �xi� yi� ∈E�G� for all i, and (b) if �xi� yj� and �xj� yk� are in E�G� and i, j, k, are distinct,then �xi� yk� ∈ E�G�.

The following characterization is interesting since it uses the language of k-covers.

Theorem 2.8. A bipartite graph is unmixed if and only if MG spans G andMG = ⋃s

i=1 Kbi�bi.

Proof. If MG spans G, by Lemma 2.6 the basic 1-covers of G are basic 1-coversof MG. By Remark 2.3, we have that each minimal cover of MG has cardinality

s∑i=1

bi�

Therefore, G is unmixed.

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If a bipartite graph is unmixed, by Villarreal’s characterization thereexists a labelling �x1� � � � � xn� y1� � � � � yn� of the vertices such that we have��x1� y1�� � � � � �xn� yn�� ⊂ E�G�. By Corollary 2.2 of [3] each minimal covering of G is

�xi1 � � � � xik � yik+1� � � � � yin��

Therefore, ��x1� y1�� � � � � �xn� yn�� ⊂ E�MG�, that is, MG spans G.The assertion follows by observing that the only complete bipartite graphs

having a perfect matching have the form Kbi�bi. �

In the article [5], Herzog and Hibi gave the first combinatoric characterizationof bipartite Cohen–Macaulay graph. In [3], Crupi, Rinaldo, and Terai generalizedthis result to edge ideals whose height is half of the number of vertices.

We specialize Proposition 3.3 and Theorem 3.4 of [3] to bipartite graphs in thefollowing way.

Lemma 2.9. Let G be a bipartite unmixed graph. Then G is Cohen–Macaulay if andonly if G has a unique perfect matching.

Theorem 2.10. A bipartite graph is Cohen–Macaulay if and only if MG spans G andMG = ⋃s

i=1 K1�1.

Proof. If G is Cohen–Macaulay, then G is unmixed and by Theorem 2.8 MG spansG and

MG =s⋃

i=1

Kbi�bi�

By Lemma 2.9, there exists only one perfect matching. Therefore, E�MG� =��x1� y1�� � � � � �xn� yn��.

Suppose that MG = ⋃si=1 K1�1; then by Theorem 2.8, G is unmixed. We may

assume that G is bipartite on the disjoint vertex sets �x1� � � � � xn� and �y1� � � � � yn�with �x1� y1�� � � � � �xn� yn� ∈ E�G�. If we show that E�MG� = ��x1� y1�� � � � � �xn� yn��is the unique perfect matching, by Lemma 2.9 we obtain the assertion.

Suppose that there exists another perfect matching, M ′. Since this matching isdifferent from E�MG�, there exists an edge, that we call e1, such that �xi1� yi2� withi1 �= i2.

We observe that xi2 belongs to another edge, say e2. Since e2 = �xi2� yi3� is in aperfect matching i2 �= i3. If i3 = i1, then �xi1� yi2� xi2� yi1� is a cycle. Otherwise, i3 �= i1,in which case there exists another edge e3 = �xi3� yi4� with i4 �= i3. Then either i4 ∈�i1� i2�, that is, there exists a 4-cycle or 6-cycle, or we continue this procedure. Sincethe graph is finite and because we have a perfect matching, eventually we end upwith a cycle that we may assume has i1 within its vertices

�xi1� yi2� xi2� � � � � xir � yir+1= yi1��

Since �xir � yi1� ∈ E�G�\E�MG�, there exists g ∈ ��G� such that g�xir �+ g�yi1� =2, that is, g�xir � = g�yi1� = 1. Since g�xij �+ g�yij � = 1 for all j = 1� � � � � r, g�xik�+

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g�yik+1� ≥ 1 for all k = 1� � � � � r − 1 and g�yi1� = 1, we obtain g�xi1� = g�xi2� = · · · =

g�xir � = 0, but this contradicts the fact that g�xir � = 1. �

3. VERTEX COVER ALGEBRA AND BIPARTITE GRAPHS

Let � = �x1� � � � � xn� be the maximal irrelevant ideal of S, and let J be theAlexander dual of the edge ideal I . Then

J = ⋂�i�j�∈E�G�

�xi� xj��

We observe that

J =( ∏

j∈Ci

xj Ci ∈ ��G�� i = 1� � � � � r)

with ��G� = �C1� � � � � Cr� the set of minimal vertex covers of G.The vertex cover algebra of G is the algebra

A�G� = ⊕k≥0

A�G�k ⊂ S�t��

where the vector space A�G�k is generated by

�xa11 · · · xann tk a is a k-cover of G��

This algebra is standard graded if and only if G is bipartite (see Theorem 5.1.(b),[7]) namely, each k-cover with k ≥ 2 is decomposable.

In this case the vertex cover algebra A�G� corresponds to the Rees algebraof J . The fiber cone of the ideal J is the algebra

�A�G� = A�G�/�A�G�

and it is standard graded, too.

Remark 3.1. A K-vector basis of �A�G�k is

�xa11 · · · xann tk a is a basic k-cover of G��

In particular, each basic k-cover is decomposable into k basic 1-covers.

By these observations, it is possible to see the strong connection between thesealgebras and combinatorics of G.

We can rephrase the nice result obtained by Benedetti, Costantinescu, andVarbaro in the article [1], Theorem 1.6, in the following way.

Theorem 3.2. Let G be a bipartite graph. Then the following are equivalent:

(1) MG spans G;(2) �A�G� is a domain.

We also recall Lemma 1.7 in the article [1].

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Lemma 3.3. Let �i� j� be an edge of a bipartite graph G. The following areequivalent:

(1) �i� j� is a nice matching;(2) If �i� i′�, �j� j′� ∈ E�G�, then �i′� j′� ∈ E�G�.

Definition 3.4. Let G be a bipartite graph with �i� j�� �i� j′� ∈ E�G�. We define anew graph Ojj′�G� with

V�Ojj′�G�� = V�G�\�j′�and

E�Ojj′�G�� = E�G�\��j′� l� ∈ E�G�� ∪ ��j� l� �j′� l� ∈ E�G���

Example 3.5. In the picture we show an example (see Definition 3.4) of a graphG and the graph Ojj′�G�:

Lemma 3.6. Let G be a bipartite graph and �i� j�� �i� j′� ∈ E�MG�. Then

�A�G� � �A�Ojj′�G���

Proof. Let G′ = Ojj′�G� and V�G� = �n�. We may assume j = n− 1, j′ = n, andV�G′� = �n− 1�. We prove that there is a bijection between the set of basic k-covers of G and the set of basic k-covers of G′ defined by

�g� = g′

with g = �g�1�� � � � � g�n��, g′ = �g�1�� � � � � g�n− 1��, k ≥ 1.By Lemma 2.7, we have that for each basic k-cover g of G

g�j� = g�n� = k− g�i��

hence ��basic k-cover of G�� = ���b��b ∈ �basic k-cover of G���.We claim that, if b is a basic k-cover of G, then �b� is a basic k-cover for G′.

To prove the claim, we observe that for a k-cover a of G being not basic implies thatthere exists l ∈ V�G� whose graph induced by the closed neighborhood of l, G�NG�l�

,with edges

��l� l1�� �l� l1�� � � � � �l� lr���

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satisfies the inequality a�l�+ a�li� > k, for i = 1� � � � � r.Let W = �n− 1� n� ⊂ V�G� (resp., W ′ = �n� ⊂ V�G′�) and its closed

neighborhood NG�W� (resp., NG′ �W ′�).We observe that the induced subgraphs G�V�G�\NG�W� and G′�V�G′�\NG�W

′� areisomorphic with respect to the labelling assigned.

Therefore, we may restrict our attention on the vertices in NG�W� (resp.,NG′ �W ′�) and their neighbors.

If l ∈ NG�W�, then �l� n− 1�� �l� n� ∈ E�G�. In fact, we may assume that l isadjacent to n− 1 and since by hypothesis �i� n− 1� is a nice matching and �i� n� isin E�G� we have by Lemma 3.3 that also �l� n� ∈ E�G�.

Therefore, for each vertex l in NG�W�, we obtain a subgraph whose edgeslexicographically ordered are

��l� l1�� � � � � �l� lr���

with lr−1 = n− 1, lr = n.By the same argument the vertex l exists also in NG′�W ′� and the corresponding

subgraph is

��l� l1�� � � � � �l� lr−1���

Now since a�l�+ a�lr−1� = a�l�+ a�lr� the k-cover a is basic with respect to l in Gif and only if it is basic with respect to l in G′.

To complete the proof of the claim, we observe that the induced subgraphsG�NG�n−1� and G�NG�n�

are isomorphic. In fact, by Lemma 3.3 �l� n− 1� ∈ E�G� if andonly if �l� n� ∈ E�G�. The claim follows since a�n� = a�n− 1�.

Therefore, we may assume �A�G� generated by xg1 t� � � � � xgr t with gi ∈��G� and �A�G′� generated by xg′1 t� � � � � xg′r t with g′i ∈ ��G′� and g′i = �gi�, fori= 1� � � � � r.

We define the graded homomorphism of K-algebras

�A�G� →�A�G′�

��xgi t��xgj t�� = �xg′i t��xg′j t�. Since in each degree k ≥ 1, the map induces abijection between the K-vector spaces �A�G�k and �A�G′�k, the assertion follows. �

Theorem 3.7. Let G be a bipartite graph such that MG spans G. Then �A�G� � �A�G′�where G′ is a bipartite Cohen–Macaulay graph.

Proof. To prove the assertion is sufficient to apply the operator defined in 3.4to each pair of adjacent edges of each connected component Km�n of MG. At theend, we obtain MG′ = ⋃s

i=1 K1�1. Hence the assertion follows by Lemma 3.6 andTheorem 2.10. �

Remark 3.8. Let G�J� = �uC1� � � � � uCr

� the minimal set of generator of the ideal J .When G is a bipartite unmixed graph, Herzog, Hibi and Ohsugi in [6], described the

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semigroup ring �G generated by the monomials uC ∈ G�J�. We observe that whenG is a bipartite unmixed graph

�A�G� � �G�

Corollary 3.9. Let G be a bipartite graph. If �A�G� is a domain, then �A�G� is Koszul.

Proof. Since �A�G� � �A�G′�, where G′ is Cohen–Macaulay, then G′ is unmixed and

�A�G′� � �G′ �

The assertion follows by Theorem 3.1 of [6]. �

Remark 3.10. In [6] the authors gave a nice characterization of Cohen–Macaulaybipartite graphs related with full sublattice of Boolean lattice (see Theorem 2.2). Byour observation (Theorem 3.7) the semigroup algebra �G, where G is a bipartiteunmixed graph, is always related with a full sublattice of a boolean lattice inducedby the Cohen–Macaulay bipartite graph G′.

Note Added at Proofs Stage

After this paper was submitted the author learned that there is an overlappingwith the recent paper of Benedetti and Varbaro (see Benedetti and Varbaro, 2011;see following article in print issue). In particular: 1) the Definition 2.1 of the graphG0=1 (Benedetti and Varbaro) that is our graph MG in Definition 2.1; 2) Theorem 2.8(.4 and .6, Benedetti and Varbaro) that is analogous with our result in Theorem 2.8.

ACKNOWLEDGMENTS

The author wishes to thank Prof. Jürgen Herzog for useful advice andsuggestions. He also wants to thank Prof. Herzog and the University of Essen forthe hospitality during his stay.

The author would like to thank the referee for helpful hints and remarks. Theauthor was partially supported by GNSAGA of INdAM (Italy).

REFERENCES

[1] Benedetti, B., Constantinescu, A., Varbaro, M. (2009). Dimension, depth andzero-divisors of the algebra of basic k-covers of a graph. Le Matematiche, Catania43:117–156.

[2] Benedetti, B., Varbaro, M. (2011). Unmixed graphs that are domains. Comm. in Algebra39:2260–2267.

[3] Crupi, M., Rinaldo, G., Terai, N. Cohen–Macaulay edge ideal whose height is halfof the number of vertices. To be published in Nagoya Mathematical Journal, Vol. 201(2011).

[4] Dupont, L. A., Villarreal, R. H. (2009). Symbolic Rees algebras, vertex covers andirreducible representations of Rees cones. Algebra Discrete Mathematics. To appear,2010.

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[5] Herzog, J., Hibi, T. (2005). Distributive lattices, bipartite graphs and Alexanderduality. J. Alg. Combin. 22:289–302.

[6] Herzog, J., Hibi, T., Ohsugi, H. (2009). Unmixed bipartite graphs and sublattices ofthe Boolean lattices. J. Alg. Combin. 30(4):415–420.

[7] Herzog, J., Hibi, T., Trung, N. V. (2007). Symbolic powers of monomial ideals andvertex cover algebras. Adv. in Math. 210:304–322.

[8] Ravindra, G. (1977). Well-covered graphs. J. Combinatorics Information Syst. Sci.2(1):20–21.

[9] Stanley, R. P. (1996). Combinatorics and Commutative Algebra. 2nd ed. Boston/Basel/Stuttgart: Birkäuser.

[10] Villarreal, R. H. (2001). Monomial Algebras. Pure and applied Mathematics.New York/Basel: Marcel Dekker.

[11] Villarreal, R. H. (2007). Unmixed bipartite graphs. Rev. Colombiana Mat. 41(2):393–395.

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