OF NON-SQUAREFREE GRAPH IDEALS
MAURIZIO IMBESI and MONICA LA BARBIERA
Communicated by the former editorial board
We investigate Cohen-Macaulay and unmixed bipartite graphs without isolated vertices, and show that graphs constructed from them by putting appropriate pairs of loops, the so-called quasi-bipartite graphs, retain such properties.
AMS 2010 Subject Classification: 05C75, 05C90, 13C14, 13H10.
Key words: Bipartite graphs with loops, vertex cover, Cohen-Macaulay and unmixed graphs.
1. INTRODUCTION
In this article we consider bipartite graphs with no isolated vertices, and expose situations for which algebraic properties of them remain when we add loops in any of their vertex.
LetGbe a graph onnverticesv1, . . . , vn. An algebraic object attached to Gis the edge idealIpGq, a monomial ideal of the polynomial ring innvariables RKrX1, . . . , Xns,K a field.
WhenG is a loopless graph,IpGqis generated by squarefree monomials of degree two in R, IpGq XiXj | tvi, vju is an edge of G
, but ifG is a graph having loopstvi, viu, among the generators ofIpGqthere are also non-squarefree monomials Xi2, i1, . . . , n.
A subset C of the vertex set tv1, . . . , vnu of a simple graph G is said a minimal vertex cover ofG if: (i) every edge ofG is incident with one vertex in C, and (ii) no proper subset ofC has the first property. IfCsatisfies condition (i) only, it is called a vertex cover of G.
Remember that a graph G isbipartite if its vertex set can be partitioned into disjoint subsetsV1 andV2 such that every edge ofGjoins V1 withV2, and the pair pV1, V2q is called abipartition ofG.
A graph G is said to be Cohen-Macaulay over the field K if the depth and the dimension overRofR{IpGqcoincide,unmixed if all its minimal vertex covers have the same cardinality, and that a Cohen-Macaulay graph is unmixed.
MATH. REPORTS15(65),2(2013), 107–113
Structural aspects of Cohen-Macaulay and unmixed bipartite graphs were studied in [1, 7]. For Cohen-Macaulay or unmixed bipartite graphs without isolated vertices we show under what conditions the graphs having loops asso- ciated to them, the so-called quasi-bipartite graphs, preserve such properties.
In particular, we introduce the definitions ofquasi-bipartitegraph, namely a bipartite graph with loops, and of strong quasi-bipartite graph, that is a complete bipartite graph having loops in every of its vertex. After recalling the combinatorial descriptions given by Herzog and Hibi in [2] and Villarreal in [7] for Cohen-Macaulay and unmixed bipartite graphs respectively, we state that a strong quasi-bipartite graph is Cohen-Macaulay, then unmixed, and we characterize the quasi-bipartite Cohen-Macaulay and unmixed graphs having the corresponding bipartite graphs such properties.
2. COHEN-MACAULAY AND UNMIXED QUASI-BIPARTITE GRAPHS
In this section we present two main results about the behaviour of Cohen- Macaulay and unmixed quasi-bipartite graphs without isolated vertices, when, for the corresponding bipartite graphs, such properties hold.
Let’s introduce some preliminary notions.
Let G be a graph and VpGq tv1, . . . , vnu be the set of its vertices.
We put EpGq ttvi, vju| vi vj, vi, vj P VpGqu the set of edges of G and LpGq ttvi, viu| vi P VpGqu the set of loops of G. Hence, tvi, vju is an edge joiningvi tovj andtvi, viuis a loop of the vertexvi. SetWpGq EpGqYLpGq. If LpGq H, the graphG is said simple or loopless, otherwise, ifLpGq H,G is a graph with loops.
A graph G on nvertices v1, . . . , vn is complete if there exists an edge for all pairs tvi, vju of vertices ofG. It is denoted by Kn.
If VpGq tv1, . . . , vnu and R KrX1, . . . , Xns is the polynomial ring over a fieldK such that each variableXi corresponds to the vertexvi, theedge ideal IpGq associated to G is the ideal XiXj| tvi, vju PWpGq
R.
Note that the non-zero edge ideals are those generated by squarefree monomials of degree 2. This implies that IpGq is a graded ideal of S of initial degree 2, that isIpGq `i¥2pIpGqiq. IfWpGq H, thenIpGq p0q.
LetG be a graph andIpGq R be its edge ideal. Letmbe the irrelevant maximal ideal of R. The depth of R{IpGq, denoted by depthpR{IpGqq, is the largest integer r for which there is an homogeneous sequence f1, . . . , fr in m such that fi is not a zero-divisor of R{pIpGq, f1, . . . , fi1q, for all 1 ¤ i ¤ r.
Such r is bounded by dimpR{IpGqq, where dimpR{IpGqq denotes the Krull dimension ofR{IpGq.
Definition 2.1.G is said to be a Cohen-Macaulay graph over K (C-M graph for short) if depthpR{IpGqq dimpR{IpGqq.
Remark 2.1.The edge idealIpGqis Cohen-Macaulay if and only ifR{IpGq is Cohen-Macaulay, i.e., depthpR{IpGqq dimpR{IpGqq.
Let GpIpGqq be the unique minimal set of monomial generators of the edge idealIpGq. A vertex cover ofIpGqis a subsetCoftX1, . . . , Xnusuch that each u PGpIpGqq is divided by some Xi PC. Such a vertex cover C is called minimal if no proper subset of C is a vertex cover ofIpGq.
Let hpIpGqqdenote the minimal cardinality of the vertex covers ofIpGq. Proposition 2.1. Let G be a graph with loops on the vertex set VpGq tv1, . . . , vnu. Let IpGq S KrX1, . . . , Xns be the edge ideal of G such that hpIpGqq n. Then G is a Cohen-Macaulay graph.
Proof. LethpIpGqqn, then it follows that dimpS{IpGqqnhpIpGqq0.
Being depthpS{IpGqq ¤dimpS{IpGqq 0. Hence,G is Cohen-Macaulay. l We are interested to study the Cohen-Macaulay property for bipartite graphs with loops.
Definition 2.2.A graph G with loops is said to be quasi-bipartite if its vertex set V can be partitioned into disjoint subsets V1 tx1, . . . , xnu and V2 ty1, . . . , ymu, every edge joins a vertex of V1 with a vertex of V2, and there exists some vertex ofV with a loop ([4]).
Definition 2.3.A graph Gwith loops is called strong quasi-bipartite if all the vertices ofV1 are joined to any vertex ofV2 and for each vertex ofV there is a loop ([4]).
The quasi-bipartite graphs determine edge ideals in two sets of variables X1, . . . , Xm;Y1, . . . , Yn, wherem is the number of the verticesx1, . . . , xm and nthe number of the verticesy1, . . . , yn. More precisely, ifGis a quasi-bipartite graph, then its edge ideal in RKrX1, . . . , Xm;Y1, . . . , Yns is
IpGq Xl2, XiYj, Yk2 | txi, yju,txl, xlu,tyk, yku PWpGq .
The following criterion by Herzog and Hibi describes all Cohen-Macaulay bipartite graphs in combinatorial terms.
Theorem 2.1 ([2], Theorem 3.4). Let G be a bipartite graph without iso- lated vertices. Then G is Cohen-Macaulay if and only if there is a bipartition V1 tx1, . . . , xgu,V2 ty1, . . . , ygu of G such that:
(a) txi, yiu PEpGq, for alli1, . . . , g;
(b) if txi, yiu PEpGq, then i¤j;
(c) if txi, yju andtxj, ykuare inEpGqand i, j, k are distinct, then txi, yku P EpGq.
For quasi-bipartite graphs the following result holds.
Corollary 2.1. A quasi-bipartite graph with a loop in each vertex is Cohen-Macaulay.
Proof. Let G be a quasi-bipartite graph with a loop in each vertex and IpGq R KrX1, . . . , Xm;Y1, . . . , Yns be its edge ideal. Being hpIpGqq m n, the thesis follows by Proposition 2.1. l
Starting from a Cohen-Macaulay bipartite graphG we will show that the property to be Cohen-Macaulay for G is preserved when we add pairs of loops to it.
Theorem 2.2. Let G be a Cohen-Macaulay bipartite graph without iso- lated vertices onV1 tx1, . . . , xgu,V2 ty1, . . . , ygu. IfGpis the quasi-bipartite graph with Vp pGq VpGq, Ep pGq EpGq and Lp pGq ttxi1, xi1u,txi2, xi2u, . . . , txis, xisu,tyi1, yi1u,tyi2, yi2u, . . . ,tyis, yisu | ti1, . . . , isu t1, . . . , gu u, then Gpis Cohen-Macaulay.
Proof. If|Lp pGq| 2g, the thesis follows by Corollary 2.1.
Now let |Lp pGq| 2g2t, forg¡t¥1.
Let t 1. |Lp pGq| 2g2 and suppose that txg, xgu,tyg, ygu R Lp pGq. The proof is by induction ong. Forg2, it is eitherIp pGq pX1Y1, X2Y2, X12, Y12q orIp pGq pX1Y1, X2Y2, X1Y2, X12, Y12q. Hence,Gpis Cohen-Macaulay. Forg¡2, letS txr1, . . . , xrpube the set of all vertices ofGpadjacent toyg. Consider the subgraphG1 pGzSobtained fromGpby removing the vertices inS. The vertices yr1, . . . , yrp are isolated vertices ofG1. In fact, if there is an edgetxi, yrjuinG1, with i rj, then by Theorem 2.1 one gets that txi, ygu P Ep pGq and xi must be a vertex in S. This is a contradiction. Using the induction hypothesis on g, the graphsG1ztyr1, . . . , yrpu and Gpztxg, ygu are Cohen-Macaulay. It follows that Gpis Cohen-Macaulay ([6], Proposition 6.2.2).
Lett¡1, theng¡2. Suppose thattxgt 1, xgt 1u,tygt 1, ygt 1u, , txg, xgu,tyg, ygu R Lp pGq and use induction on g. For g 3, Ip pGq can be generated bytX1Y1, X2Y2, X3Y3, X12, Y12uortX1Y1, X2Y2, X3Y3, X1Y2, X12, Y12u or tX1Y1, X2Y2, X3Y3, X1Y3, X12, Y12u or tX1Y1, X2Y2, X3Y3, X2Y3, X12, Y12u or tX1Y1, X2Y2, X3Y3, X1Y2, X2Y3, X1Y3, X12, Y12u.
Hence, Gpis Cohen-Macaulay. Now suppose g¡3. Let S1 txl1, . . . , xlqu be the set of all vertices of Gp adjacent to ygt 1 and consider the subgraph
G2 pGzS1. As shown in the first part of the proof, the vertices yl1, . . . , ylq of G2 are isolated. By induction hypothesis on g, the graphs G2ztyl1, . . . , ylqu andGpztxgt 1, ygt 1uare Cohen-Macaulay, and this implies thatGpis Cohen- Macaulay. l
Now, we study the unmixed property for quasi-bipartite graphs.
Recall that a subset C of VpGq is said to be a minimal vertex cover of G if every edge of G is incident with one vertex in C and there is no proper subset of C having such a property. Notice that C is a minimal vertex cover if and only if VpGqzC is a maximal independent set. The smallest number of vertices in any minimal vertex cover of G is called vertex covering number.
Definition 2.4.A graph G is called unmixed if all the minimal vertex covers of G have the same number of elements.
The notion of unmixed graph is related to some other theoretical and al- gebraic graph properties. The implication Cohen-Macaulayñunmixed holds for any graph without isolated vertices [6].
Minimal vertex covers of unmixed bipartite graphs G with no isolated vertices were first described in [2]. Iftx1, . . . , xmuYty1, . . . , ynudenotes the set of vertices of G, sincetx1, . . . , xmu and ty1, . . . , ynu represent minimal vertex covers of G, then m n. Moreover, by the marriage theorem, there is a perfect matching for G, so it may be assumed that txi, yiu is an edge of G, for all i, and following this fact, each minimal vertex cover ofG is of the form txi1, . . . , xis, yis 1, . . . , yinu, where ti1, . . . , inu t1, . . . , nu.
Later Villarreal gave the following combinatorial characterization of all unmixed bipartite graphs with no isolated vertices.
Theorem 2.3 ([7], Theorem 1.1). Let G be a bipartite graph without isolated vertices. Then G is unmixed if and only if there is a bipartition V1 tx1, . . . , xgu,V2 ty1, . . . , ygu of G such that:
(a) txi, yiu PEpGq, for alli1, . . . , g;
(b) if txi, yju andtxj, yku are in EpGq and i, j, k are distinct, then txi, yku PEpGq.
Starting from an unmixed bipartite graph G we will show that the prop- erty to be unmixed for G is preserved when we add pairs of loops to it.
Theorem 2.4. Let G be an unmixed bipartite graph without isolated ver- tices on V1 tx1, . . . , xgu, V2 ty1, . . . , ygu. If Gpis the quasi-bipartite graph
withVp pGqVpGq,Ep pGq EpGqandLp pGq ttxi1, xi1u,txi2, xi2u,. . . ,txis, xisu, tyi1, yi1u,tyi2, yi2u, . . . ,tyis, yisu | ti1, . . . , isu t1, . . . , gu u, then Gpis unmixed.
Proof. For|Lp pGq| 2g, the assertion holds by Corollary 2.1 remembering that Cohen-Macaulay implies unmixed.
If |Lp pGq| 2pgtq, for t 1, . . . , g1, let G be the subgraph of Gp obtained by removing all vertices of Gphaving loops. Thus, G is a bipartite subgraph of G with 2t non isolated vertices, xi1, . . . , xit PV1 and yi1, . . . , yit P V2, such thattxij, yiju PEpGq, forij 1, . . . , t, and whentxih, yiku,txik, yilu P EpGq with ih ik il, then txih, yilu P EpGq. By Theorem 2.3, G is unmixed and, according to the proof of [7], Theorem 2.1, any minimal vertex cover for it intersects every edge txij, yijuin exactly one vertex.
On the other hand, let G denote the subgraph of Gp whose edge set is given by Ep pGqzEpGq. We state that G is also unmixed. In fact, any vertex cover for it must contain at least all the 2pgtq vertices with loops of G. Byp construction, every other vertex of G is adjacent to some of such vertices, so a minimal vertex cover for G has cardinality 2pgtq.
Note that the graph Gpis the union of the graphs G and G.
For showing that Gp is unmixed, it is enough to prove that all minimal vertex covers for it have 2gt vertices.
If G and G are disjoint, a minimal vertex cover for Gp is just the union set of a minimal vertex cover for G and one for G. When G and G are not disjoint, there exist vertices with no loop common to such subgraphs. These vertices are ends of that edges belonging to G whose endpoints are not both with loops. The adding of these edges toG doesn’t increase the cardinality of any minimal vertex cover for G itself, considered as a subgraph ofG, becausep a vertex cover for G must intersect the vertices with the loop of such edges.
So again, any minimal vertex cover for Gpis the union set of a minimal vertex cover for G and one forG, and it is formed by 2pgtq tvertices. l
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Received 30 April 2011 University of Messina, Department of Mathematics, Viale Ferdinando Stagno d’Alcontres, 31
98166 Messina, Italy imbesim@unime.it University of Messina, Department of Mathematics, Viale Ferdinando Stagno d’Alcontres, 31
98166 Messina, Italy monicalb@dipmat.unime.it
