In a paper in 2008, Herzog, Hibi and Ohsugi introduced and studied the
semigroup ring associated to the set of minimal vertex covers of an unmixed
bipartite graph. In this paper we relate the dimension of this semigroup ring
to the rank of the Boolean lattice associated to the graph.Comment: 6 pages, Pragmatic 2008, University of Catania (Italy); corrected
  typo

Bertone, Cristina

Micale, Vincenzo

English

arXiv

In a paper in 2008, Herzog, Hibi and Ohsugi introduced and studied the semigroup ring associated to the set of minimal vertex covers of an unmixed bipartite graph. In this paper we relate the dimension of this semigroup ring to the rank of the Boolean lattice associated to the graph

Le Matematiche (Dipartimento di Matematica e Informatica, Università degli Studi di Catania)

LE MATEMATICHEVol. LXIII (2008) – Fasc. II, pp. 157–163ON THE DIMENSION OF THE MINIMAL VERTEX COVERSEMIGROUP RING OF AN UNMIXED BIPARTITE GRAPHCRISTINA BERTONE - VINCENZO MICALEIn a paper in 2008, Herzog, Hibi and Ohsugi introduced and studiedthe semigroup ring associated to the set of minimal vertex covers of anunmixed bipartite graph. In this paper we relate the dimension of thissemigroup ring to the rank of the Boolean lattice associated to the graph.1. IntroductionLet G be a finite graph without loops, multiple edges and isolated vertices andletM (G) be the set of minimal vertex covers of G. In [2, Section 3] the authorsintroduce and study the semigroup ring associated to the minimal vertex coversof an unmixed and bipartite graph G.In this paper we relate the dimension of this semigroup ring to the rank ofthe Boolean lattice associated to G.In Section 2, we recall the concept of an unmixed bipartite graph G and wegive some preliminaries about the Boolean lattice associated to G. In particular,we characterize those sublattices of the Boolean lattice which are associatedto G (cf. Theorem 2.3). Then, in the particular case of bipartite graphs, weconcentrate on the concept of vertex cover algebra.In Section 3 we define the semigroup ring associated to the minimal vertexcovers of an unmixed and bipartite graph G and we prove that its dimensionEntrato in redazione: 2 gennaio 2009AMS 2000 Subject Classification: 13P10Keywords: Unmixed graph, bipartite graph, vertex cover algebra158 CRISTINA BERTONE - VINCENZO MICALEequals the rank of LG plus one (cf. Theorem 3.3). As a particular case of thisresult we get that the dimension of the semigroup ring associated to the minimalvertex covers of bipartite and Cohen-Macaulay graphs on 2n vertices is equal ton+1 (cf. Corollary 3.4).2. PreliminariesThroughout this paper, graphs are assumed to be finite, loopless, without multi-ple edges and isolated vertices. We denote by V (G) the set of vertices of G andby E(G) the set of edges of G.Definition 1. For a graph G, a subset C of the set of vertices V (G) is called avertex cover for G if every edge of E(G) is incident to at least one vertex fromC. C is a minimal vertex cover if for any C′ (C, C′ is not a vertex cover for G.LetM (G) denote the set of minimal vertex covers of G. In general, the minimalvertex covers of a graph do not have the same cardinality.Example 2.1. Let G be the graph with V (G) = {1,2,3,4,5} andE(G) = {{1,2},{2,3},{3,4},{1,4},{4,5}}.ThenM (G) = {{2,4},{1,3,5}}.Definition 2. A graph G is unmixed if all the elements ofM (G) have the samecardinality.Example 2.2. The graph G with V (G) = {1,2,3,4} and E(G) = {{1,2},{2,3},{3,4},{1,4}} is unmixed asM (G) = {{2,4},{1,3}}.Definition 3. A graph G is bipartite if its set of vertices V (G) can be dividedin two disjoint subsets U and V such that, for all l ∈ E(G), we have |l ∩U | =1 = |l∩V |.In what follows G will be assumed to be bipartite and unmixed with respectto the partition V (G) =U ∪V of its vertices, where U = {x1, . . . ,xm} and V ={y1, . . . ,yn}.Since G is unmixed and U and V are both minimal vertex cover for G, thenn = m.Furthermore, let U ′ ⊆U and N(U ′) be the set of those vertices y j ∈V for whichthere exist a vertex xi ∈ U ′ such that {xi,y j} ∈ E(G); then (cf.[1, p. 300]),since (U \U ′)∪N(U ′) is a vertex cover of G for all subset U ′ of U and sinceG is unmixed, it follows that |U ′| ≤ |N(U ′)| for all subset U ′ of U . Thus, theON THE DIMENSION OF THE MINIMAL VERTEX COVER... 159marriage theorem enable us to assume that {xi,yi} ∈ E(G) for i = 1, . . . ,n.We can also assume that each minimal vertex cover of G is of the form{xi1 , . . . ,xis ,yis+1 , . . . ,yin}where {i1, . . . , in}= [n] = {1, . . . ,n}.For a minimal vertex cover C = {xi1 , . . . ,xis ,yis+1 , . . . ,yin} of G, we set C′ ={xi1 , . . . ,xis}. LetLn denote the Boolean lattice of all the subset of {x1, . . . ,xn}and letLG = {C′ |C is a minimal vertex cover of G }. One easily checks this isa sublattice ofLn. SinceLn is a distributive lattice, any sublattice is distributiveas well.Actually, there is a one to one correspondence between the graphs we arestudying and the sublattices ofLn containing /0 and {x1, . . . ,xn}:Theorem 2.3. [2, Theorem 1.2] Let L be a subset of Ln. Then there existsa (unique) unmixed bipartite graph G on {x1, . . . ,xn} ∪ {y1, . . . ,yn} such thatL =LG if and only if /0 and {x1, . . . ,xn} belong toL andL is a sublattice ofLn.2.1. Cohen-Macaulay bipartite graphsLet A be the polynomial ring K[z1, . . . ,zN ] over a field K. To any graph G onvertex set [N], let I(G) be the ideal of A, called the edge ideal of G, generatedby the quadratic monomials ziz j such that {i, j} ∈ E(G).Definition 4. A graph G is Cohen-Macaulay if the quotient ring A/I(G) isCohen-Macaulay.Let, as before,Ln denote the Boolean sublattice on {x1, . . . ,xn}.Definition 5. The rank of a sublattice L of Ln, rankL , is the non-negativeinteger l where l + 1 is the maximal cardinality of a chain of L . A sublatticeL ofLn is called full if rankL = n.Theorem 2.4. [2, Theorem 2.2] A subset L of Ln is a full sublattice of Ln ifand only if there exists a Cohen-Macaulay bipartite graph G on {x1, . . . ,xn}∪{y1, . . . ,yn} withL =LG.2.2. Vertex cover algebraLet G be a bipartite and unmixed graphs on the set of vertices {x1, . . . ,xn} ∪{y1, . . . ,yn} and with minimal vertex cover C = {xi1 , . . . ,xis ,yis+1 , . . . ,yin}. It isuseful to notice that xi ∈C if and only if yi /∈C.We can identify C with the (0,1)-vector, bC ∈ N2n such that160 CRISTINA BERTONE - VINCENZO MICALEbC( j) =1 if 1≤ j ≤ n and x j ∈C1 if n+1≤ j ≤ 2n and y j−n ∈C0 otherwisewhere bC( j) denotes the j-th coordinate of the vector bC.In this way, we can associate to each minimal vertex cover C of G a square-free monomial in the polynomial ring S = K[x1, . . . ,xn,y1, . . . ,yn] with degxi =degyi = 1; in fact, we first associate to C its vector bC and then we consider themonomial uC = xbC(1)1 · · ·xbC(n)n ybC(n+1)1 · · ·ybC(2n)n .Definition 6. The vertex cover algebra of the bipartite graph G is the subal-gebra A(G) of S[t] generated, over S, by the monomials uCt for every minimalvertex cover C of G, that is A(G) = S[uCt, C ∈M (G)].By [3, Theorem 4.2 and Corollary 4.4], we have, in particular, that A(G) is afinitely generated, graded, normal, Gorenstein S-algebra.Moreover, in [3, Theorem 5.1], the authors show that A(G) is generated indegree ≤ 2 and that it is standard graded.3. The dimension of A(G)We now introduce the object of our study in this paper.Let m be the maximal graded ideal of S. For a bipartite unmixed graph G,we consider the standard graded K-algebraA(G) := A(G)/mA(G)∼= K[uCt, C ∈M (G)]∼= K[uC, C ∈M (G)].Hence A(G) is the semigroup ring generated by all monomials uC such thatC ∈M (G). This object has been introduced and studied in [2, Section 3], wherethe authors proved, in particular, that A(G) is a normal and Koszul semigroupring (cf. [2, Corollary 3.2]).The aim of the paper is to relate the dimension of A(G) to rankLG (cf.Theorem 3.3).Let d = |M (G)| and BG be the d× 2n matrix whose rows are exactly thevectors bC. Since C1 = {x1, . . . ,xn} and C2 = {y1, . . . ,yn} are always inM (G),we can assume that the first and the last rows of BG are bC1 and bC2 respectively.Finally, let b˜C be the n-vector containing only the first n entries of bC and let B˜Gbe the d×n matrix whose rows are the vectors b˜C.Lemma 3.1. rankBG = rank B˜G+1.ON THE DIMENSION OF THE MINIMAL VERTEX COVER... 161Proof. Let C1, . . . ,C2n be the column vectors of BG (note that the columns of B˜Gare exactly C1, . . . ,Cn) and let C˜ be the column vector with d entries each equalsto 1. Since Cn+ j = C˜−C j for every j = 1, . . . ,n, then we have that〈C1, . . . ,Cn,C˜〉K = 〈C1, . . . ,Cn,Cn+1, . . . ,Cn〉K .as K-vector spaces.Finally, since the last entry in each column C1, . . . ,Cn is 0, it follows thatC˜ /∈ 〈C1, . . . ,Cn〉K , that isdimK〈C1, . . . ,Cn,C˜〉K = dim〈C1, . . . ,Cn〉K +1.Lemma 3.2. rank B˜G = rankLGProof. Let rankLG = m and consider a chain of maximal length m+ 1 in thesublattice LG. We note that, by Theorem 2.3, /0 and [n] are in this chain. Eachelement of this maximal chain corresponds to a row of the matrix B˜G. Let denotewith v1, . . . ,vm+1 the row vectors associated to this maximal chain, where v1 isthe vector associated to the element at the top of the chain, v2 is the vectorassociated to the element of the chain just below the top, and so on for theremaining vectors v3, . . . ,vm+1. With this notation we have that v1 is the vectorwith all 1’s and vm+1 is the vector with all 0’s. We note that if i > j, then thenumbers of 1’s in vi is strictly less than the number of 1’s in v j and that if 0 isthe l-th coordinate of vi, then 0 is the l-th coordinate of v j. This two facts implythat v1, . . . ,vm are linearly indipendent. So we have rank B˜G ≥ m.In order to prove that equality holds, we show that all the other rows of B˜Gare linear combination of the m rows associated to v1, . . . ,vm. With an abuse ofnotation, we now identify the elements of the lattice LG with their associatedvectors. SinceLG is a lattice containing [n] and /0, following the maximal chainin the lattice containing the vectors v1, . . . ,vm, we have, at a certain height, thesituation depicted in the picture...........................................................................................................................................................................................vivi+1vi+2v162 CRISTINA BERTONE - VINCENZO MICALEwhere vi,vi+1,vi+2 are in the maximal chain.But LG is a distributive lattice and, in terms of the vectors, this means thatwe can obtain v from the other vectors in the picture: in fact (vectorially)v = vi− vi+1+ vi+2.Repeating this in each analogous situation, we have that all the possiblevectors representing elements of the lattice which are not in the chosen maximalchain, can be obtained by a linear combination of the vectors v1, . . . ,vm. In termsof the matrix B˜G, this means that rank B˜G ≤ m.Theorem 3.3. Let G be an unmixed, bipartite graph on 2n vertices with noisolated vertices and letLG be the associated sublattice ofLn. Thendim A(G) = rankLG+1.Proof. By [4, Proposition 7.1.17], we have that dim A(G) = rank BG. By Lem-mas 3.1 and 3.2, we get the proof.Corollary 3.4. Let G be a Cohen-Macaulay bipartite graph on 2n vertices.Thendim A(G) = n+1Proof. By [4, Proposition 6.1.21], G is unmixed. Furthermore, by Theorem 2.4,Cohen-Macaulay graphs correspond to full sublattices. Hence, by Theorem 3.3,we get the thesis.AcknowledgementsThe authors wish to thank Professors Ju¨rgen Herzog and Volkmar Welker for thevaluable conversations concerning this paper. We also wish to thanks ProfessorAlfio Ragusa and all other organizers of PRAGMATIC 2008 for the opportunityto partecipate and for the pleasant atmosphere they provided during the summerschool.ON THE DIMENSION OF THE MINIMAL VERTEX COVER... 163REFERENCES[1] J. Herzog - T. Hibi, Distributive Lattices, Bipartite Graphs and Alexander Duality,Journal of Algebraic Combinatorics 22 (2005), 289–302.[2] J. Herzog - T. Hibi - H. Ohsugi, Unmixed bipartite graphs and sublattices of theBoolean lattices, Preprint, 2008.[3] J. Herzog - T. Hibi - N. V. Trung, Symbolic powers of monomial ideals and vertexcover algebras, Advances in Mathematics 210 (1) (2007), 304–322.[4] R. H. Villarreal, Monomial Algebras, Marcel Dekker, 2001.CRISTINA BERTONEDipartimento di MatematicaUniversita` di Torinoe-mail: cristina.bertone@unito.itVINCENZO MICALEDipartimento di Matematica e InformaticaUniversita` di Cataniae-mail: vmicale@dmi.unict.it

On the dimension of the minimal vertex cover semigroup ring of an unmixed bipartite graph

In a paper in 2008, Herzog, Hibi and Ohsugi introduced and studied the semigroup ring associated to the set of minimal vertex covers of an unmixed bipartite graph. In this paper we relate the dimension of this semigroup ring to the rank of the Boolean lattice associated to the graph.</p

Cristina Bertone

Vincenzo Micale

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On the dimension of the minimal vertex covers semigroup ring of an unmixed bipartite graph

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