Discrete Mathematics 302 (2005) 107–123www.elsevier.com/locate/disc

Multiplicities of edge subrings

Isidoro Gitler1, Carlos E. ValenciaDepartamento de Matemáticas, Centro de Investigación y de Estudios Avanzados del IPN,

Apartado Postal 14–740, 07000 México City, D.F., Mexico

Received 21 November 2002; received in revised form 4 July 2003; accepted 22 July 2004Available online 9 August 2005

Abstract

For a bipartite graphG we are able to characterize the complete intersection property of the edgesubring in terms of the multiplicity and we give optimal bounds for this number. We give a methodto obtain a regular sequence for the atomic ideal ofG, whenG is embedded on an orientable surface.We also give a graph theoretical condition for the edge subring associated withG to be Gorenstein.Finally we give a formula for the multiplicity of edge subrings, of arbitrary simple graphs.© 2005 Elsevier B.V. All rights reserved.

MSC:Primary 13H10; Secondary 13F20, 13P10

Keywords:Multiplicity; Edge subrings; Complete intersection; Bipartite graphs

1. Introduction

LetG= (V ,E) be a bipartite graph with vertex setV = {v1, . . . , vn}. Consider a ring ofpolynomialsR=K[x1, . . . , xn]=⊕∞

i=0Ri over a fieldKwith the standard grading inducedby deg(xi) = 1. Theedge subringof G is theK-subalgebra:

K[G] := K[{xixj | vi is adjacent tovj }] ⊂ R,

E-mail address:igitler@math.cinvestav.mx(I. Gitler).1 Partially supported by CONACyT grant 40201-F and SNI.

0012-365X/$ - see front matter © 2005 Elsevier B.V. All rights reserved.doi:10.1016/j.disc.2004.07.029

108 I. Gitler, Carlos E. Valencia / Discrete Mathematics 302 (2005) 107–123

whereK[G] is given thenormalized gradingK[G]i =K[G]∩R2i . ThusK[G] is a normalCohen–Macaulay standardK-algebra[16,20]. The Hilbert series ofK[G] is the rationalfunction given by

F(K[G], t) =∞∑i=0

dimK(K[G]i )t i .

Thea-invariant ofK[G] is the degree of the Hilbert series ofK[G] as a rational functionand is denoted bya(K[G]). TheHilbert polynomialof K[G] is the unique polynomial�(t) ∈ Q[t] such that

dimK(K[G]i ) = �(i) = ad−1id−1 + ad−2i

d−2 + · · · + a1i + a0, (i?0)

whered is the Krull dimension ofK[G]. Themultiplicity of K[G], denoted bye(K[G]),is the integerad−1(d − 1)!. If G is a spanning subgraph of the complete bipartite graphKm,n−m with m<n, then the following is a sharp bound for the multiplicity:

e(K[G])�e(K[Km,n−m]) =(

n − 2

m − 1

),

see[6]. It is easy to produce examples where this bound is very loose. The extreme casebeing whenG is a tree, because in this case themultiplicity ofK[G] is equal to 1. In Section2 we observe that the family of connected bipartite planar graphs without cut vertices admitsharp bounds.We use this bound to characterize the complete intersection property of edgesubrings of bipartite graphs. This property was studied in[8], and later in[17,19] somecharacterizations were shown.We give amethod to obtain a regular sequence for the atomicideal of agraphG, whenG is bipartite andembeddedonaorientable surface, ageneralizationof the results obtained in[8]. In Section 3 we examine the canonical module ofK[G] andobtain a graph-theoretical condition for the ringK[G] to be Gorenstein.In Section 4 we study the multiplicity of the edge subring of an arbitrary simple graph

G. LetA= {�1, . . . , �q} be the set of all vectors�k = ei + ej ∈ Rn such thatvi is adjacentto vj , whereei is theith unit vector inRn. Consider the edge polytope

P = conv(�1, . . . , �q) ⊂ Rn,

where conv(A) is the convex hull ofA. TheEhrhart ringA(P ) ofP is the gradedK-algebradefined as

A(P ) = K[{x�t i | � ∈ Zn ∩ iP }] ⊂ R[t],wheret is a new variable. We present a formula relating the multiplicities ofK[G] andA(P ), see Theorem 4.9. Basic references for terminology and notation on Ehrhart rings andgraphs are[3,11,14,22,27].

2. Bounding the multiplicity

In this sectionG will denote a graph with vertex setV (G) = {v1, . . . , vn} and edge setE(G) = {z1, . . . , zq}.

I. Gitler, Carlos E. Valencia / Discrete Mathematics 302 (2005) 107–123 109

A vertexv of a connected graphG is acut vertexif G\{v} is disconnected. A graphG is2-connectedif G is connected, hasmore than two vertices andGhas no cut vertices.A blockof G is either a maximal 2-connected subgraph or an edge (bridge). By their maximality,different blocks ofG intersect in at most one vertex, which is then a cut vertex ofG.Therefore every edge ofG lies in a unique block andG is the union of its blocks[7].Thetoric idealP(G)of the edge subringK[G] is the kernel of the gradedhomomorphism

of K-algebras

S = K[t1, . . . , tq ] −→ K[G]induced by the assignmentti �→ fi , where{f1, . . . , fq} is the set of all monomialsxixjsuch thatvi is adjacent tovj andS is a polynomial ring with the standard grading.A sequence� = �1, . . . , �s in the ringR is called a regular sequence of anR-moduleM

if (�)M �= M and

�i /∈Z(M/(�1, . . . , �i−1)M) for all 2� i�s,

where

Z(N) = {x ∈ R | xn = 0 for some 0�= n ∈ N}is the set of zero divisors ofN and(B) is the ideal generated by the setB. The ringK[G]is called acomplete intersectionif P(G) is generated by a homogeneous regular sequence.By [25] the toric ideal is generated by binomials corresponding to cycles:

P(G) = ({tc | c is a cycle inG}),where

tc = ti1ti3 · · · ti2r−1 − ti2ti4 · · · ti2r ,for c = ti1, . . . , ti2r an even closed walk.If G can be written asG = G1 ∪ G2, whereG1 andG2 are subgraphs with at most one

vertex in common, then

K[G]�K[G1]⊗KK[G2] and P(G) = (P (G1), P (G2)),

moreover

a(K[G]) = a(K[G1]) + a(K[G2])and

type(K[G]) = type(K[G1]) + type(K[G2]),where type(K[G]) is the last Betti number in the minimal free resolution of the represen-tationR/I of K[G] as anR-module.ThereforeK[G] is a complete intersection (resp. Gorenstein) if and only ifK[G1] and

K[G2] are complete intersections (resp. Gorenstein). Hence we may assume thatG is 2-connected.

110 I. Gitler, Carlos E. Valencia / Discrete Mathematics 302 (2005) 107–123

Definition 2.1. A chordof a cyclec in a graphG is any edge ofG joining two nonadjacentvertices ofc. A cycle without chords is calledprimitive.

Lemma 2.2(Doering and Gunston[8] ). If d is a cycle of G andtd ∈ ({tc | c is a cycle,c �= d}), then d has a chord.

Proposition 2.3. The toric idealP(G) is a prime ideal of S of heightq − n+ 1minimallygenerated by the set{tc | c is a primitive cycle ofG}.

Proof. As dim(K[G]) = n − 1, the result is a consequence of Lemma 2.2 and the fact thatP(G) is generated by binomials that correspond to the cycles ofG. �

In what follows we will consider bipartite graphs embedded on orientable surfaces. Wealways assume minimum genus embeddings. Furthermore we assume that all our embed-dings are simple, that is, the regions induced by the embedding are open 2-cells boundedby simple circuits (closed walks with no repeated vertices or edges).We refer to the regionsas faces of the embedding. We will denote the set of faces of the embedding byF(G). If cis a closed walk that bounds a face we say thatc is anatomic cycle.The idealA(G) ⊂ P(G) generated by

A(G) = ({tc | c is an atomic cycle inG}),is called the atomic ideal.

Definition 2.4. Let c be an even closed walk ofG, c = ti1, . . . , tis . A polarizationof cconsists in choosing one of the sets

{ti1, ti3, . . . , tis−1} or {ti2, ti4, . . . , tis }.Let {c1, . . . , cr} be the set of atomic cycles ofG. A polarizationP of G is a set

P = {P1, . . . ,Pr}wherePi is a polarization ofci for all i,Pi ∩ Pj = ∅ for i �= j and

r⋃i=1

Pi = E(G).

The next result is a generalization to nonplanar graphs of the planar case given in[8,Proposition 2.2].

Theorem 2.5. LetG= (V ,E) a bipartite graph simply embedded on an orientable surfaceS, then G has a polarization.

Proof. LetGbe a graph with vertex setV (G)={v1, . . . , vn}, edge setE(G)={z1, . . . , zq}and face setF(G), embedded on the surfaceS. Letm(G) be the medial graph ofG withvertex setV (m(G)) = {u1, . . . , uq} [see,[11]]. Recall thatm(G) is a 4-regular graph. By

I. Gitler, Carlos E. Valencia / Discrete Mathematics 302 (2005) 107–123 111

construction,m(G) is also cellularly embedded onS. In particular the faces ofm(G),F(m(G)), can be 2-colored, where one color class corresponds to the set of vertices ofG,denoted byF1 and the other class corresponds to the faces ofG, denoted byF2. We colorwith black the faces inF1 and with white the faces inF2.Let us construct a new graphm′(G) fromm(G). Forui ∈ V (m(G)), let {yi

1, yi2, y

i3, y

i4}

be the vertices adjacent toui . Splitui into two new verticesu′i andu

′′i , so thatu

′i is adjacent

to yi1 andy

i2, u

′′i is adjacent toy

i3 andy

i4 and add a new edge betweenu′

i andu′′i , as shown

in the following figure.

Repeat the process for all vertices ofm(G) to obtainm′(G).It is easy to see that the graphm′(G) is bipartite by showing that all its cycles are even.

Since any cycle can be obtained as the symmetric difference of atomic cycles, we onlyconsider these cycles. LetCbe an atomic cycle, ifC is inF1, then it corresponds to a vertexv ∈ V (G); the cycle associated to this vertex has, by construction, a length twice the degreeof v in G, so it is always even. Now ifC is in F2 then it corresponds to a cycle ofF(G).SinceG is bipartite, then the cycle is also even, because in the splitting process no vertex isadded to the boundary of a white face.Sincem′(G) is bipartite, we take a 2-coloring of its vertices. A polarization is obtained

by fixing one of these color classes and considering the set of vertices in this class for everyatomic cycle inF2. �

We illustrate with an example the constructions given in the proof.

Example 2.6. LetG = K4,4 and consider the following embedding ofG in the torusT.Fig. 1 showsK4,4 embedded in the torusT; Fig. 2 shows the medial graph ofK4,4

in the torus; inFig. 3 the color classes are represented by the white vertices and the blackvertices and inFig. 4 the polarization is obtained by taking pairs of edges with lines in theinterior of each atomic cycle.From this embedding we obtain the following polarization

P =

P1 = {t1, t8}, P2 = {t2, t9}P3 = {t3, t10}, P4 = {t4, t11}P5 = {t5, t12}, P6 = {t6, t13}P7 = {t7, t14}, P8 = {t15, t16}

,

112 I. Gitler, Carlos E. Valencia / Discrete Mathematics 302 (2005) 107–123

v1 v2

v1 v2

v3 v4

v5

v7

v5

v7

v6

v8

c1 c6

c4 c7

c5

c3

c3

c2 c2

c8

c8

c8

c8

Fig. 1.K4,4 embedded inT.

u1

u5

u9

u13

u2

u6

u10

u14

u3

u7

u11

u15

u4

u8

u12

u16

c1 c6

c4 c7

c5

c3

c3

c2 c2

c8

c8

c8

c8

Fig. 2. The medial graphm(K4,4).

where

t1 = {x1, x5}, t2 = {x3, x5}, t3 = {x1, x6}, t4 = {x3, x7},t5 = {x3, x8}, t6 = {x2, x6}, t7 = {x4, x8}, t8 = {x3, x6},t9 = {x4, x7}, t10 = {x2, x8}, t11= {x1, x8}, t12 = {x4, x6},t13= {x4, x5}, t14= {x2, x9}, t15= {x1, x7}, t16= {x2, x5},

as is shown inFig. 4.

I. Gitler, Carlos E. Valencia / Discrete Mathematics 302 (2005) 107–123 113

c1 c6

c4 c7

c5

c3

c3

c2c2

c8

c8

c8

c8

Fig. 3.m′(K4,4) with a 2-coloring of its vertices.

v1 v2

v1 v2

v3 v4

v5

v7

v5

v7

v6

v8

c1 c6

c4 c7

c5

c3

c3

c2 c2

c8

c8

c8

c8

Fig. 4. The polarization ofK4,4.

Lemma 2.7. Let G be a bipartite graph simply embedded on an orientable surfaceS and{c1, . . . , cr} the atomic cycles of G. IfP = {P1, . . . ,Pr} is a polarization of G, then byrelabeling the edges one may assume that

ti ∈ Pi for all i = 1, . . . , r − 1

andtj /∈ ci if 1� i < j �r − 1.

114 I. Gitler, Carlos E. Valencia / Discrete Mathematics 302 (2005) 107–123

Proof. We use induction onr, the number of atomic cycles. Ifr −1=1 the lemma is clear.Take an edget in the atomic cyclecr such thatt /∈Pr , label this edgetr−1. The edgetr−1appears in an atomic cycle distinct fromcr , that we labelcr−1, and is contained inPr−1.LetG′ be the graph obtained fromG by deleting all the edges and vertices that appear inthe atomic cyclescr−1 andcr . Recall thatG′ hasr − 1 atomic cycles{c1, . . . , cr−2, c

′r−1},

wherec′r−1 is the atomic cycle obtained fromcr−1 andcr when the edges and vertices in

cr−1 ∩ cr are deleted. Apply the induction hypothesis toG′, with the polarization inducedbyG, labeling witht1 throughtr−2, to finish. �

Proposition 2.8. Let G be a bipartite graph simply embedded on an orientable surfaceS.If c1, . . . , cr are the atomic cycles of the graph G, then

tc1, . . . , t̂ci , . . . , tcr ,

wheretc1, . . . , t̂ci , . . . , tcr means{tc1, . . . , tcr }\{tci } for some0� i�r, is a regular sequencefor A(G) with respect to a certain lexicographic ordering.

Proof. Exchange the labels of the atomic cyclesci andcr . Assume we have a labeling ofthe edges as in Lemma 2.7. Let> be the lexicographic ordering onSwith

t1> t2> · · ·> tr−1> · · ·> tq ,

let in(f ) be the initial term of the polynomialf with respect to this ordering and let in(I ) bethe ideal generated by the initial term of the polynomials inI with respect to this ordering.Hence

mi = in(tci ) =∏

{tk ∈ Pi}.We have that gcd(mi,mj )= 1 for all i �= j , becausePi ∩Pj = ∅. Thereforetc1, . . . , tcr−1

is a regular sequence and the height of(tc1, . . . , tcr−1) is given by:

ht((tc1, . . . , tcr−1)) = ht(in((tc1, . . . , tcr−1)))

= ht(m1, . . . , mr−1) = r − 1. �

We have the following conjecture.

Conjecture 2.9. If c1, . . . , cr are the atomic cycles of the graph G, then

tc1, . . . , t̂ci , . . . , tcr

is a Gröbner basis forA(G) with respect to a certain lexicographic ordering.

Example 2.10.LetK4,4 be embedded in the torusT as in Example 2.6. Then consideringany of the subsets with seven binomials from the set

R =

t1t8 − t2t3, t2t9 − t4t13t3t10 − t6t11, t4t11− t5t15t5t12 − t7t8, t6t13− t12t16t7t14− t9t10, t15t16− t1t14

we obtain a regular sequence forA(G).

I. Gitler, Carlos E. Valencia / Discrete Mathematics 302 (2005) 107–123 115

Recall that a graphG is planar if and only if it can be embedded in the sphere. The casefor planar graphs was previously given in[8].

Corollary 2.11 (Doering and Gunston[8] ). Let G be a bipartite planar graph, then G hasa polarization.

Proof. It follows from Theorem 2.5. �

Proposition 2.12. Let ri denote the number of atomic cycles of length2i in a bipartiteplanar graph G, then

e(K[G])�2r23r3 · · · srs ,is a sharp bound for the multiplicity ofK[G].

Proof. First observe thatr=r1+· · ·+rs is the height ofP(G), since the height isq−n+1and this number is the dimension of the cycle space ofG, see[14,25]. There is a gradedepimorphism ofK-algebras:

S/A(G) −→ S/P (G) � K[G].Since the Hilbert polynomials ofS/A(P ) andS/P (G) have the same degree we obtainthat e(A(G)) is an upper bound for the multiplicity ofK[G]. As S/A(G) is a completeintersection its Hilbert series is given by

F(S/A(G), z) =∏r

i=1 (1+ z + · · · + zai−1)

(1− z)n−1 ,

whereai is the degree of the binomialtci associated to the atomic cycleci . Hence makingz = 1 in the numerator givese(A(G)) = 2r23r3 · · · srs .To construct a planar graph with multiplicitye(A(G)), recall thatP(G) is a complete

intersection if and only if any two cycles inG with no chord have at most one edge incommon, see[17]. Thus any connected bipartite planar graphG without cut vertices suchthatP(G) is a complete intersection would reach this bound provided it hasri atomic cyclesof length 2i for eachi. Such a graph can be obtained starting with a cycle of length 2i andgluing ri cycles of this length one by one, where at each step the new cycle added sharesexactly one edge with the previous graph.�

Theorem 2.13.Let G be a bipartite planar graph and letri denote the number of atomiccycles in G of length2i. Then

e(K[G]) = 2r23r3 · · · srs

if and only ifA(G) = P(G).

Proof. If A(G) = P(G), thenP(G) is a complete intersection and the required formulafor the multiplicity follows from the proof of Proposition 2.12.

116 I. Gitler, Carlos E. Valencia / Discrete Mathematics 302 (2005) 107–123

Conversely assume the equalitye(K[G]) = 2r23r3 · · · srs . Let c1, . . . , cr be the atomiccycles ofG. By [8] there is a monomial order≺ of Ssuch thattc1, . . . , tcr is a Gröbner basisforA(G), and bothtc1, . . . , tcr and its leading terms sequence lt(tc1), . . . , lt(tcr ) are regularsequences. By[27, Proposition 8.1.10]the set

{tc | c is a cycle inG}is a universal Gröbner basis ofP(G). Thus in(P (G)), the initial ideal ofP(G)with respectto ≺, is a square-free monomial ideal. From the proof of[23, Proposition 13.15]and us-ing that a shellable simplicial complex is Cohen–Macaulay it follows thatS/in(P (G)) isCohen–Macaulay. Note that

e(S/in(P (G))) = e(S/P (G)) = e(S/A(G)) = e(S/in(A(G))).

Since themultiplicitye(S/in(P (G))) (resp.e(S/in(A(G))) is thenumberofminimal primesof in(P (G)) (resp. in(A(G))) and all those primes have the same heightr, using primarydecompositions it follows readily that

in(A(G)) = (lt(tc1), . . . , lt(tcr )) = in(P (G)).

HenceP(G) = (tc1, . . . , tcr ) = A(G) as required. �

Corollary 2.14. Let ri be the number of atomic cycles in G of length 2i. If

e(k[G]) = 2r23r3 · · · srs ,then k[G] is a complete intersection.

3. A condition for the Gorenstein property

Let G be a bipartite simple graph with vertex setV = {v1, . . . , vn}. Consider the setA= {�1, . . . , �q} of all the vectors�k = ei + ej ∈ Rn such thatvi is adjacent tovj , whereei is theith unit vector inRn. Theedge coneR+A of G is defined as thecone generatedbyA, that is,R+A is the set of all the linear combinations of�1, . . . , �q with nonnegativecoefficients. HereR+ denotes the set of nonnegative real numbers. NoteR+A �= (0) if Ghas at least one edge. By[13] one has

n − c0(G) = dimR+A,

wherec0(G) is the number of components ofG.If a ∈ Rn, a �= 0, then the setHa will denote thehyperplaneof Rn through the origin

with normal vectora, that is,

Ha = {x ∈ Rn | 〈x, a〉 = 0}.This hyperplane determines twoclosed half-spaces

H+a = {x ∈ Rn | 〈x, a〉�0} andH−

a = {x ∈ Rn | 〈x, a〉�0}.A subsetF ⊂ Rn is a faceof the edge cone ofG if there is asupporting hyperplaneHa such that (i)F = R+A ∩ Ha �= ∅, and (ii) R+A ⊂ H−

a or R+A ⊂ H+a . If in

I. Gitler, Carlos E. Valencia / Discrete Mathematics 302 (2005) 107–123 117

additionR+A /⊂ Ha we callF aproper faceandHa aproper supporting hyperplane. Theempty set andR+A are theimproper faces. A proper faceF of the edge cone is afacetifdim F = dim R+A − 1.

Lemma 3.1. If vi is not an isolated vertex of G, then the setF = Hei ∩ R+A is a properface of the edge cone.

Proof. NoteF �= ∅ because 0∈ F , andR+A ⊂ H+ei. Sincevi is not an isolated vertex

R+A /⊂ Hei . �

Let A be an independent set of vertices ofG, that is, no two vertices ofA are adjacentand letN(A) be its neighbor set, that is, the set of vertices ofG adjacent to some vertex inA. The supporting hyperplane of the edge cone defined by∑

vi∈Axi =

∑vi∈N(A)

xi

will be denoted byHA. The sum over an empty set is defined to be zero.

Theorem 3.2(Valencia and Villarreal[24]). If G is connected graph with bipartition(V1, V2), then there is a unique irreducible representation

R+A = aff(R+A) ∩(

u⋂i=1

H−Ai

)∩(⋂

i∈IH+

ei

)

such thatAi�V1 for all i , I = {i | vi ∈ V2 is not cut vertex} and none of the closedhalfspaces can be omitted from the intersection. Hereaff(R+A) is the affine hull ofR+A.

Definition 3.3. An integral matrixA is totally unimodularif all the i × i minors ofA areequal to 0 or±1 for all i.

Lemma 3.4. Zn ∩ R+A = NA.

Proof. LetM be the matrix whose columns are the vectors inA = {�1, . . . , �q}. Take� ∈Zn ∩ R+A, then by Carathéodory’s Theorem[10, Theorem 2.3]and after an appropriatepermutation of the�i ’s we can write

� = �1�1 + · · · + �r�r (�i �0),

wherer is the rank ofM and�1, . . . , �r are linearly independent. On the other hand sincethe submatrixM ′ = (�1 · · · �r ) is totally unimodular becauseG is bipartite, by Kronecker’slemma[18, p. 51]the system of equationsM ′x = � has an integral solution. Hence� is alinear combination of�1, . . . , �r with coefficients inZ. It follows that�i ∈ N for all i, thatis, � ∈ NA. The other containment is clear.�

Corollary 3.5. Let G be a connected graph with bipartition(V1, V2)and let�=(�1, . . . , �n)

an integral vector inR+A, then∑n

i=1 �i is an even integer.

118 I. Gitler, Carlos E. Valencia / Discrete Mathematics 302 (2005) 107–123

Proof. By Theorem 3.2 we have thatR+A ⊂ aff(R+A) and aff(R+A) = Hd withd =∑i∈Vi

ei −∑j∈V2 ej . Now, Since� ∈ R+A, then∑n

i=1 �i =∑

i∈Vi�i +

∑j∈V2 �j =

2∑

i∈V1 �i . �

Proposition 3.6. K[G] is a normal domain and its canonical module is the ideal ofK[G]given by

�K[G] = ({xa | a ∈ Zn ∩ ri(R+A)}) ⊂ K[G], (1)

whereri(R+A) is the relative interior of the edge cone.

Proof. It follows using Lemma 3.4 and the Danilov–Stanley formula for the canonicalmodule, see[3]. �

The followingmembership criterion is the analogous of[26, Proposition 3.4]for bipartitegraphs.

Proposition 3.7. A monomialx�11 · · · x�n

n of R belongs toK[G] if and only if�=(�1, . . . , �n)

is an integral vector of the edge cone of G.

Proof. It follows by Lemma 3.4. �

Remark 3.8. Let G be a graph without isolated vertices and take a vector� in ri(R+A).If F is a proper face ofR+A, then� /∈F , see[1, Theorems 5.3]. In particular�i >0 for alli, becauseHei ∩ R+A is a proper face.

Recall thatK[G] is Gorensteinif its canonical module is a principal ideal. The nexttheorem is a surprising combinatorial obstruction forK[G] to be Gorenstein. SinceK[G]is Gorenstein if and only ifK[Gi] is Gorenstein for each blockGi of G, then it suffices toconsider only the case whenG is a 2-connected bipartite graph.

Theorem 3.9. Let G be a Gorenstein2-connected bipartite graph with bipartition(V1, V2),then G has a perfect matching and furthermore

|A|< |N(A)| for all A�V1.

Proof. Let x� be the generator of�K[G]. For each 1� i�m+p choose a spanning tree ofG\{vi} and enlarge this to a spanning treeTi of G. Note that the monomialx�i with

�i = (degTi(v1), . . . ,degTi

(vi)︸ ︷︷ ︸1

, . . . ,degTi(vp+m))

is the generator of the canonical module ofK[Ti]. Hence since aff(R+Ai ) = aff(R+A),whereR+Ai is theedgeconeofTi ,wegetx�i ∈ �K[G].Therefore�i=1, andconsequently� = 1= (1, . . . ,1). Noting that� is in the affine space generated by the edge cone ofG itfollowsm = p. Thus from Theorem 3.2 we obtain|A|< |N(A)| for all A�V1. �

I. Gitler, Carlos E. Valencia / Discrete Mathematics 302 (2005) 107–123 119

Example 3.10.Consider the bipartite graphG

v1 v8

v5

v7

v2

v4

v6

v3

with edge ideal generated by the monomials

x1x5, x1x6, x1x7, x2x5, x2x6, x2x8, x3x6, x3x7, x3x8, x4x7, x4x8.

Using Macaulay2[12] the ringK[G] has type three. Since(1, . . . ,1) is in the relativeinterior of the edge cone, the converse of Theorem 3.9 does not hold.

4. Ehrhart rings and multiplicities

LetG be a graph with vertex setV = {x1, . . . , xn} and letA= {�1, . . . , �q} be the set ofvectors inNn of the formei + ej such thatxi is adjacent toxj . Recall that theincidencematrix of G, denoted byA, is just the matrix with column vectors�1, . . . , �q . Consider apolynomial ringR = K[x1, . . . , xn] over a fieldK. Theedge polytopeof G is the latticepolytope

P = conv(�1, . . . , �q) ⊂ Rn,

where conv(A) is the convex hull ofA. TheEhrhart ringA(P ) of P is defined as

A(P ) = K[{x�t i | � ∈ Zn ∩ iP }] ⊂ R[t],

wheret is a new variable. The subring generated by the monomials ofA(P ) of t-degreeequal to 1 is called thepolytopal subringand is denoted byK[P ]. Thus

K[P ] = K[{x�t | � ∈ Zn ∩ P }] ⊂ R[t],

120 I. Gitler, Carlos E. Valencia / Discrete Mathematics 302 (2005) 107–123

this ring is studied in[2,15]. The Ehrhart ringA(P ) is a finitely generatedK-algebra and itis a gradedK-algebra withith component given by

A(P )i =∑

�∈Zn∩iP

Kx�t i .

One of the properties ofA(P ) is that its Hilbert function:

E(i) = |Zn ∩ iP | = cdid + · · · + c1i + c0 (i?0)

is a polynomial function of degreed = dim(P ) such thatd!cd is an integer, which is themultiplicity of A(P ). The rational polynomial

EP (x) = cdxd + cd−1x

d−1 + · · · + c1x + c0

is called theEhrhart polynomialof P. By [22] the relative volume ofP is:

vol(P ) = limi→∞

|Zn ∩ iP |id

.

Hence vol(P ) is the leading coefficient of the Ehrhart polynomial ofP. For this reasond!cdis called thenormalized volumeof P. In the sequel vol(P ) will denote the relative volumeof a lattice polytopeP as defined in[22]. Note thatA(P ) andEP (x) can be computedusing[4,5].

Lemma 4.1. K[Gt] = K[{x�1t, . . . , x�q t}] = K[P ].

Proof. It follows from the equality conv(�1, . . . , �q) ∩ Zn = {�1, . . . , �q}. �

LetA be the incidence matrix ofG. Recall that dimK[G] is equal to rank(A) = n − c0and dimK[P ] is equal to dim(P ) + 1, wherec0 is the number of connected bipartitecomponents ofG, see[27]. On the other hand by Lemma 4.1K[G] � K[P ], and conse-quently dim(P ) = n − c0 − 1.We define the matrixB by

B =(

�1 · · · �q1 · · · 1

),

where�1, . . . , �q are regarded as column vectors. Observe thatAandBhave the same rankbecause the last row ofB is a linear combination of the firstn rows.If A is an integralmatrix,�i (A)will denote the greatest commondivisor of all the nonzero

i × i minors ofA. A graph is unicyclic if it has only one cycle.

Lemma 4.2. If G is a unicyclic graph with its unique cycle having odd length andr = rank(B), then�r (B) = 1.

Proof. If G is an odd cycle of lengthn, then it is not hard to see that the matrixB ′ obtainedby deleting any of the firstn rows ofB has determinant±1.

I. Gitler, Carlos E. Valencia / Discrete Mathematics 302 (2005) 107–123 121

We proceed by induction. IfG is not an odd cycle, thenG has a vertexxj of degree 1.Thus thejth row ofA has exactly one entry equal to 1, sayajk = 1. Consider the matrixC obtained fromB by deleting thejth row and thekth column. Thus by induction we have�r−1(C) = 1, which gives�r (B) = 1. �

Lemma 4.3. Let G be a graph with connected componentsG0, . . . ,Gm. If G0 is nonbi-partite, then there exists a subgraph H of G with connected componentsH0, . . . , Hm suchthatH0 is a spanning nonbipartite unicyclic subgraph ofG0 andHi is a spanning tree ofGi for all i�2.

Proof. The existence ofH2, . . . , Hm is clear because any connected graph has a spanningtree. Take a spanning treeT0 of G0. For each edgee of G0 not in T0, the graphT0 ∪ {e}has exactly one cycle. The setZ(T ) of such cycles forms a basis for the cycle space ofG0.Since a graph is bipartite if and only if every cycle is even, there is an edgee of G0 suchthatT0∪{e} is a unicyclic connected graph (with respect to some cycle basis) with a uniqueodd cycle. �

Proposition 4.4. If G is a graph with exactly one nonbipartite connected component andr = rank(B), then�r (B) = 1.

Proof. Let G0, . . . ,Gm be the components ofG, with G0 nonbipartite, and letH be asin Lemma 4.3. IfA′ is the incidence matrix ofH, then using thatG andH have the samenumber of nonbipartite components one has

rank(A) = rank(A′).

Therefore�r (B) = 1 by Lemma 4.2. �

If (M,+) is an abelian group, itstorsion subgroupwill be denoted byT (M).

Proposition 4.5. Let G be a nonbipartite graph and let A be its incidence matrix. If A hasrank r, then

�r (A) = 2�r (B).

Proof. SetA′ = {(�1,1), . . . , (�q,1)}. It suffices to prove that there is an exact sequenceof groups

0 −→ T (Zn+1/ZA′) �−→ T (Zn/ZA)−→ Z2 −→ 0.

Following [21] we define� and as follows. For� = (a1, . . . , an) ∈ Zn andb ∈ Z, set�(�, b) = � and(�) = a1 + · · · + an. It is not hard to verify that� is injective and thatim(�) = ker().To show that is onto consider a nonbipartite componentG1 of G. If x1, . . . , xs are

the vertices ofG1 and�′1, �

′2, . . . , �

′m are the columns of the incidence matrix ofG1, then

Zs/(�′1, �

′2, . . . , �

′m) is a finite group. It follows that for any 1�j �s, the elementej is in

the torsion subgroup ofZn/ZA, hence(ej ) = 1, as required. �

122 I. Gitler, Carlos E. Valencia / Discrete Mathematics 302 (2005) 107–123

Corollary 4.6 (Grossman et al.[13]). If G is a graph andc0 (resp.c1) is the number ofbipartite (resp. nonbipartite) components, then

Zn/ZA � Zc0 ⊕ Zc12 .

Proof. Let G1, . . . ,Gc be the components ofG and letAi be the set of vectors inAcorresponding to the edges of the subgraphGi . Since the vectors inAi can be regarded asvector inZ�i , where�i is the number of vertices inGi , there is a canonical isomorphism

Zn/ZA � Z�1/ZA1 ⊕ · · · ⊕ Z�c /ZAc.

To finish the proof apply Propositions 4.4 and 4.5 to derive

Z�i /ZAi �{

Z2 if Gi is non bipartite,Z if Gi is bipartite.

In the second isomorphism we are using the fact that incidence matrices of bipartite graphsare totally unimodular (see[18, Chapter 19]). �

Corollary 4.7. If c1 is the number of nonbipartite components of G and r is the rank of B,then

�r (B) ={2c1−1 if c1�1,1 if c1 = 0.

Proof. It follows from Proposition 4.5 and Corollary 4.6.�

We state the following result to be used below.

Theorem 4.8(Escobar[9] ). If A={�1, . . . , �q} ⊂ Zn is a set of vectors lying on an affinehyperplane not containing the origin andP = conv(A), then

vol(P ) = |T (Zn/(�2 − �1, . . . , �q − �1))| limi→∞

|ZA ∩ iP |id

,

whered = dim(P ) andT (M) denotes the torsion subgroup of M.

The following result reduces the calculation of thea-invariant of the Ehrhart ring to thecalculation of thea-invariant of the monomial subring ofG. Giving an interesting relationbetween both monomial subrings.

Theorem 4.9. If G is a graph and P is its edge polytope, then

e(A(P )) = vol(P )d! ={2c1−1e(K[G]) if G is nonbipartite,e(K[G]) if G is bipartite,

whered = dim(P ) andc1 is the number of nonbipartite components of G.

Proof. Using Theorem 4.8 it follows thate(A(P )) = �r (B)e(K[P ]), wherer is the rankof A. From Corollary 4.6 one derives�r (A) = |T (Zn/ZA)| = 2c1. Thus the result followsreadily from Proposition 4.5 and using the fact thatA is totally unimodular ifc1 = 0. �

I. Gitler, Carlos E. Valencia / Discrete Mathematics 302 (2005) 107–123 123

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