ON THE HILBERT SERIES OF VERTEX COVER ALGEBRAS OF UNMIXED BIPARTITE GRAPHS

ALGEBRAS OF UNMIXED BIPARTITE GRAPHS

CRISTIAN ION

We compute the reduced Gr¨obner basis of the toric ideal with respect to a suitable monomial order and we study the Hilbert series of the vertex cover algebraA(G), whereGis an unmixed bipartite graph without isolated vertices.

AMS 2010 Subject Classification: 05E40, 13P10.

Key words: unmixed bipartite graph, vertex cover algebra, toric ideal, Hilbert series.

1. INTRODUCTION

LetG= (V, E) be a simple (i.e., finite, undirected, loopless and without multiple edges) graph with the vertex set V = [n] and the edge setE =E(G).

Fork∈N, ak-vertex cover ofGis a vectorc= (c1, c2, . . . , cn)∈Nⁿ such that c_i+c_j ≥k for every edge{i, j} ofG.

Let R = K[x₁, x₂, . . . , x_n] be the polynomial ring over a field K. The vertex cover algebra A(G) is defined as the subalgebra of the one variable polynomial ring R[t] generated by all monomials x^c₁¹x^c₂². . . x^c_nⁿt^k, where c = (c₁, c₂, . . . , c_n) is ak-vertex cover of G. This algebra was introduced and stud- ied in [7]. Let mbe the maximal graded ideal of R. ¯A(G) = A(G)/mA(G) is called the basic cover algebra and it was studied in [6], [1] and [2]. In [8], the Hilbert series of A(G), H_A(G)(z), for a Cohen-Macaulay bipartite graph Gis studied and several consequences are derived.

Our main aim in this paper is to extend the study ofH_A(G)(z) to unmixed bipartite graphs. It will turn out that many of the results concerning the Cohen-Macaulay case extend naturally to the larger class of unmixed bipartite graphs. The first step in getting the formula for H_A(G)(z) is to compute the toric ideal ofA(G). This is done in Section 2.

In the last section we state the main theorem which relates H_A(G)(z) to the Hilbert series of the basic cover algebras ¯A(G_F), for allF ⊂[n]. Based on this formula we derive sharp bounds for the multiplicity of A(G).

MATH. REPORTS13(63),4 (2011), 337–346

2. THE LATTICE ASSOCIATED TO AN UNMIXED BIPARTITE GRAPH

Let G be an unmixed bipartite graph without isolated vertices. By [9, Theorem 1.1] we may assume that G admits a bipartition of its vertex set V_n = W ∪W⁰, where W = {x₁, . . . , x_n} and W⁰ = {y₁, . . . , y_n}, n ≥ 1, such that

(a){x_i, y_i} ∈E(G), for alli∈[n];

(b) if{x_i, yj} ∈E(G) and {x_j, yk} ∈E(Gn), then {x_i, yk} ∈E(Gn), for all distinct i, j, k ∈[n].

Throughout this paper, whenever we refer to an unmixed bipartite graph we assume it is given with its above bipartition.

For ∅ 6= F ⊂ [n] we denote by G_F the subgraph of G induced by the subset VF ={x_i |i∈F} ∪ {y_i|i∈F}.

Remark 2.1. GF also satisfies (a) and (b), hence GF is an unmixed bipartite graph onV_F and each minimal vertex cover ofG_F has the cardinality equal to |F|.

LetK_{i,j}, 1≤i < j≤n, be the complete bipartite graph on {x_i, x_j} ∪ {y_i, yj}.

Lemma 2.2. Let G be an unmixed bipartite graph on V_n = W ∪W⁰, n≥2. Suppose thatGhas an induced subgraph K_{i,j} with1≤i < j≤n. Let H be the subgraph of G induced by the subset Vn\{x_j, yj}. Then there exists a one-to-one correspondence between the sets M(G), respectively M(H), of minimal vertex covers of G, respectively H. More precisely, for all subsets C ⊂V_n\{x_j, y_j} we have

(i) if xi ∈C, then C ∈ M(H)⇔C∪ {x_j} ∈ M(G);

(ii)if x_i6∈C, then C∈ M(H)⇔C∪ {y_j} ∈ M(G).

Proof. LetC∈ M(H). Ifx_i ∈C, put B=C∪ {x_j}. We show thatB∈ M(G). B∩ {x_k, y_l} 6=∅, for all {x_k, y_l} ∈E(H) and B ∩ {x_j, y_l} 6=∅, for all {x_j, y_l} ∈E(G). Let {x_k, yj} ∈E(G) with k /∈ {i, j}. Since {x_k, yj} ∈ E(G) and {x_j, y_i} ∈E(G), it follows, by (b), that{x_k, y_i} ∈E(G). Hence{x_k, y_i} ∈ E(H) and C∩ {x_k, yi} 6= ∅. C ∈ M(H) implies that |C∩ {x_i, yi}| = 1 and, sincex_i ∈C, we have thaty_i ∈/C. On the other hand,C∩ {x_k, y_i} 6=∅, hence xk∈C. ThusB∩ {x_k, yi} 6=∅and B ∈ M(G). If yi ∈C, putB =C∪ {y_j}.

Similarly, it can be proved that B∈ M(G).

Conversely, letB ∈ M(G). Then |B∩ {x_j, y_j}|= 1, which implies that eitherB∩ {x_j, yj}={x_j}orB∩ {x_j, yj}={y_j}. IfB∩ {x_j, yj}={x_j}, then, sinceB∩ {x_i, y_j} 6=∅, it follows that x_i ∈B. PutC =B∩(V_n\{x_j, y_j}). For all {x_k, yl} ∈ E(H) we have ∅ 6= B ∩ {x_k, yl} ⊂ B\{x_j} = C, hence C is a vertex cover of H. Since |C|= |B| −1−1, we get C ∈ M(H). Similarly, if B∩ {x_j, yj}={y_j}, thenxj ∈/ B and C=B\{y_j} ∈ M(H).

Herzog and Hibi proved in [6, Theorem 1.2] that each unmixed bipartite graph on V_n = W ∪W⁰, n ≥ 1, can be uniquely associated to a sublat- tice L_G of the Boolean lattice L_n on {p₁, p2, . . . , pn} such that ∅ ∈ L_G and {p₁, p2, . . . , pn} ∈ L_G. The lattice L_G is defined as {α ⊂ {p₁, p2, . . . , pn} | (∃)C ∈ M(G) :p_k∈α⇔x_k ∈C}.

Corollary 2.3. In the hypothesis of Lemma 2.2 and with the above notation, we have L_H ' L_G.

Proof. Letν :L_H → L_G defined by ν(α) =

(α∪ {p_j} ifp_i∈α α ifpi6∈α. . By Lemma 2.2, ν is well defined and bijective. It can be easily checked thatν is a lattice isomorphism.

We show that G has a unique Cohen-Macaulay bipartite subgraph G⁰, up to a graph isomorphism, such that the lattices associated to Gand G⁰ are isomorphic.

Proposition2.4. Let Gbe an unmixed bipartite graph onVn=W∪W⁰, n≥1, without isolated vertices. Then there exists a Cohen-Macaulay bipartite subgraph G⁰ of G, unique up to a graph isomorphism, such that L_G⁰ ' L_G. In particular, all maximal chains of L_G have the same length.

Proof. We proceed by induction on n. If n = 1, then G is Cohen- Macaulay. Put G⁰ =G and the assertion trivially holds.

Let us suppose that n >1. If G is Cohen-Macaulay, then put G⁰ = G.

If G is not Cohen-Macaulay, then, by [4, Theorem 3.4], G has an induced subgraph K_{i,j}with 1≤i < j ≤n. Let H be the subgraph of G induced by the subsetV_n\{x_j, y_j}ofV_n. By the induction hypothesis there exists a unique Cohen-Macaulay bipartite subgraphG⁰ ofH, up to a graph isomorphism, such that L_G⁰ ' L_H. Obviously, G⁰ is also a subgraph of G. By Corollary 2.3, L_H ' L_G, hence L_G⁰ ' L_G. Since G⁰ is Cohen-Macaulay, it follows, by [6, Theorem 2.2], that L_G⁰ is a full sublattice of the Boolean lattice on{p_i|xi∈ V(G⁰)}, which implies that all maximal chains of L_G⁰ have the same length, hence the conclusion.

Remark 2.5. Let G be an unmixed bipartite graph on Vn = W ∪W⁰, n ≥ 1, without isolated vertices. One may derive a procedure to compute a Cohen-Macaulay bipartite subgraphG⁰ofGsuch that the latticesL_GandL_G⁰ are isomorphic. In fact, G⁰ = GF, where F is a maximal subset of [n] such that K_{i,j} is not an induced subgraph of G_F, for all distincti, j∈F.

Remark 2.6. If G⁰ = GF, F ⊂ [n], is a Cohen-Macaulay bipartite sub- graph of G from Proposition 2.4, then, by Corollary 2.3, the lattice isomor- phism ν : L_G⁰ → L_G is defined by ν(α⁰) = α⁰ ∪ {p_j | j ∈ [n]\F, pi ∈ α⁰, K_{i,j} ⊂G}, for all α⁰∈ L_G⁰.

3. A GR ¨OBNER BASIS OF THE TORIC IDEAL OF THE VERTEX COVER ALGEBRA OF AN UNMIXED BIPARTITE GRAPH

Let S = K[x1, . . . , xn, y1, . . . , yn] and let G be an unmixed bipartite graph on Vn =W ∪W⁰, n≥ 1, without isolated vertices. In this case A(G) is the Rees algebra of the cover ideal I_G, which is generated by all mono- mials x^c₁¹. . . x^c_nⁿy^c₁ⁿ⁺¹. . . y_n^c2n, where c = (c1, . . . , c2n) is a 1-vertex cover ofG ([7]). Thus

A(G) =S⊕I_Gt⊕ · · · ⊕I_G^kt^k⊕ · · · .

LetL_Gbe the lattice associated toG. PutBG=K[{x_i}_1≤i≤n,{y_j}_1≤j≤n, {u_α}_α∈LG]. For each α ∈ L_G we denote mα =

Q

pi∈α

xi

· Q

pj6∈α

yj

. The toric ideal Q_G of A(G) is the kernel of the surjective ring homomorphism ξ :B_G →A(G),ξ(x_i) =x_i,ξ(y_i) =y_i, 1≤i≤n,ξ(u_α) =m_αt,α∈ L_G.

Let <_lex the lexicographic order on S induced by the ordering of the variables x₁ > · · · > x_n > y₁ > · · · > y_n. Let <^# the reverse lexicographic order on the polynomial ring K[{u_α}_α∈LG] induced by an ordering of the variables u_α’s such that u_α > u_β if β ⊂ α in L_G. Let <^]_lex the monomial order on B_G defined as the product of the monomial orders<_lex and<^#from above. This monomial order was introduced in [5].

Next, inspired by [5, Theorem 1.1], we compute the reduced Gr¨obner basis of the toric ideal of the vertex cover algebra of an unmixed bipartite graph G on V_n =W ∪W⁰, n≥1, with respect to the monomial order <^]_lex. Forα∈ L_GletV(α) be the set of all upper neighbours ofαinL_G. We denote x_β\α = Q

pi∈β\α

x_i, y_β\α = Q

pi∈β\α

y_i, where α ∈ L_G, α 6= {p₁, p₂, . . . , p_n} and β ∈V(α).

Theorem 3.1. Let G be an unmixed bipartite graph on Vn =W ∪W⁰, n ≥1, without isolated vertices. Then the reduced Gr¨obner basis of the toric ideal Q_G of the vertex cover algebra A(G) with respect to <^]_lex is

G_<# lex

(QG) ={x_β\αuα−yβ\αuβ |α∈ L_G, α6={p₁, p2, . . . , pn}, β ∈V(α)}∪

∪{u_αu_β−uα∪βuα∩β |α, β∈ L_G, α6⊂β, β6⊂α},

where the initial monomial of each binomial is the first monomial.

Proof. We essentially follow the proof of [5, Theorem 1.1] with a slight modification in its last part. As it was shown there, we only need to consider a primitive binomial g of the reduced Gr¨obner basis of QG with respect to

<^]_lex. Letg∈ G_<# lex

(Q_G),

g=

n

Y

i=1

x^a_iⁱy_i^bi

!

(uα1uα2. . . uαr)−

n

Y

i=1

x^a

0 i

i y^b

0 i

i

!

u_α⁰

1u_α⁰

2. . . u_α⁰_r

, with α₁ $α₂ $. . .$α_r and α⁰₁ $α₂⁰ $. . . $α⁰_r chains inL_G and in_<#

lex

(g) equal to the first monomial of g.

We assume that g /∈ K[{u_α}_α∈LG]. As in [5, Theorem 1.1] we get that there exists some 1≤j≤r such that α⁰_j 6⊂αj. Let β be an upper neighbour of αj withαj ⊂β ⊂αj∪α⁰_j and let pi ∈β\α_j. Thenpi ∈α⁰_k for allk≥j and p_i∈/ α_lfor alll≤j. This implies thata_i >0 for allifor whichp_i ∈β\α_j. Then the binomialh=xβ\α_juαj−y_β\αjuβ ∈QGand in_<^#

lex

(h) =xβ\α_juαj|in_<^#

lex

(g).

Hence in_<# lex

(g) must coincide withx_β\αju_αj, and, moreover,h=g.

4. THE HILBERT SERIES OF VERTEX COVER ALGEBRAS OF UNMIXED BIPARTITE GRAPHS

LetG be an unmixed bipartite graph on V_n=W ∪W⁰, n≥1, without isolated vertices and let{m₁, m2, . . . , m_l}be the minimal system of generators of I_G. We viewA(G) as a standard gradedK-algebra by assigning to each x_i and yj, 1≤i, j≤n and to eachmkt, 1≤k≤l, the degree 1.

The Hilbert function and the Hilbert series of the vertex cover algebra A(G) are invariant to a certain class of graph isomorphisms.

Remark 4.1. Letσbe a permutation of [n] and let^σGdenote the bipartite graph on Vn=W ∪W⁰ with the edge setE(^σG) ={{x_σ(i), y_σ(j)} | {x_i, yj} ∈ E(G)}. The graph isomorphismh:V(G)→V(^σG),h(xi) =x_σ(i)andh(yj) = y_σ(j), i, j ∈ [n], induces a K-automorphism of S which maps IG onto I^σG. Therefore, A(G) andA(^σG) have the same Hilbert function and series.

Let ∆(L_G) be theorder complex of the lattice L_G. (We refer the reader to [3, §5.1] for the definition and properties of the order complex associated to a poset.) Let S_G =K[{u_α}_α∈LG] be the polynomial ring in |L_G|variables over K. The toric ideal ¯QG of the basic cover algebra ¯A(G) is the kernel of the surjective ring homomorphismπ :S_G→A(G) defined by¯ π(u_α) =m_α, for all α∈ L_G.

Proposition 4.2. The graded K-algebra A(G)¯ and the order complex

∆(L_G) have the same vector h-vector.

Proof. By [6, Proposition 3.1] ¯QG is a graded ideal and the initial ideal in_<#( ¯Q_G) of the toric ideal ¯Q_Gcoincides with the Stanley-Reisner idealI_∆(LG), henceS_G/Q¯_GandK[∆(L_G)] have the sameh-vector. SinceS_G/Q¯_Gand ¯A(G) are isomorphic as gradedK-algebras, the conclusion follows.

Remark 4.3. Let G be an unmixed bipartite graph on Vn = W ∪W⁰, n ≥ 1, without isolated vertices and let G⁰ be a Cohen-Macaulay bipartite subgraph of G with L_G ' L_G⁰. If h, respectively h⁰, are the h-vectors of A(G), respectively ¯¯ A(G⁰), then, by using Proposition 4.2 and the fact that the lattices L_G and L_G⁰ are isomorphic, it follows that h =h⁰. Moreover, by [8, Remark 1.3], hi ≥ 0, for all 0 ≤ i ≤ r + 1 and hr = hr+1 = 0, where r = rank(L_G).

In order to prove the main theorem we need some preparatory notation and results. They are closely related to those for the Cohen-Macaulay case which were proved in [8].

Let∅ 6=F $[n],Pn(F) ={p_i |i∈F}and letα∈ L_G¯

F, where ¯F denotes the complement set ofF in [n]. We denote byδ_αthe maximal subset ofP_n(F) such that α∪δα∈ L_G. Note that

δ_α=∪γ, where γ ⊂P_n(F) and α∪γ ∈ L_G.

If we setβ =α∪δ_α,then, by the definition ofδ_α, β has the following property:

there exists no subset ∅ 6= A ⊂ F such that β ∪ {p_i | i ∈ A} is an upper neighbour of β inL_G.

Lemma 4.4. Let ∅ 6= F $ [n] and let S be the set of all β ∈ L_G with the property that there exists no subset ∅ 6=A⊂F such that β∪ {p_i |i∈A}

is an upper neighbour of β in L(G). Then the map ϕ:L_G¯

F → S defined by α 7→β =α∪δα, is an isomorphism of posets.

Proof. We follow the proof of Lemma 1.4 in [8]. We show that ϕ is invertible. Indeed, the map ψ :S → L(G_F¯) defined by ψ(β) = β∩P_n( ¯F) is the inverse of ϕsinceα=β∩Pn( ¯F) andα∈ L(GF¯).

We prove thatϕis an isomorphism of posets. Let α₁, α₂∈ L(GF¯) with α1 $α2 andβi =ϕ(αi) =αi∪δi, i= 1,2.We only need to show thatβ1 ⊂β2

since the strict inclusion follows from the hypothesis α₁$α₂. Let us assume that β₁ 6⊂β₂ and let p_r1 ∈β₁\β₂ with r₁ ∈F. Since p_r1 ∈/ δ₂, it follows that β2∪ {p_r1}∈ L/ _G.

We claim that {u ∈ F | p_u ∈ β₁\{p_r1}} 6= ∅. Let us suppose, on the contrary, that pu 6∈ β1\{p_r1}, for all u ∈ F. Then β1 = α1 ∪ {p_r1}. Since β₁, β₂ ∈ L_G, it follows thatβ₁∪β₂∈ L_G. On the other hand, we haveβ₁∪β₂= β2∪ {p_r1}, which implies that β1∪β2 ∈ L/ _G, a contradiction.

By repeated application of this argument we get the sequencer1, r2, . . . , r_k, r_k+1, . . .withr_k+1 ∈F\{r₁, . . . , r_k}andp_rk+1 ∈β₁\(β₂∪{p_r1, . . . , p_rk}), for all k≥0. Therefore, the setF is infinite, which is impossible. Henceβ1 ⊂β2. Now let β1, β2 ∈ S with β1 $ β2 and assume that α1 = α2, where α1 =β1∩P_n( ¯F),andα2 =β2∩P_n( ¯F).Thenδ1 =β1\P_n( ¯F)$δ2 =β2\P_n( ¯F).

But this is impossible since δ1 is maximal among the subsets γ ⊂Pn(F) such that α₁∪γ ∈ L_G.

The next result relates the Hilbert series of the vertex cover algebras A(G) to the Hilbert series of the basic covers algebras ¯A(GF), for allF ⊂[n].

If F =∅, we put by conventionHA(G¯ F)(z) = _1−z¹ .

Theorem4.5. LetGbe an unmixed bipartite graph onVn=W∪W⁰,n≥ 1, without isolated vertices. ForF ⊂[n]let r_F = rank(L_GF), letHA(G¯ F)(z) be the Hilbert series ofA(G¯ _F), andH_A(G)(z)be the Hilbert series ofA(G). Then:

(1) H_A(G)(z) = 1

(1−z)ⁿ X

F⊂[n]

HA(G¯ _F)(z) z

1−z n−|F|

.

In particular, if h(z) = P

j≥0

hjz^j, respectively h^F(z) = P

j≥0

h^F_jz^j, where h= (h_j)j≥0, respectivelyh^F = (h^F_j )j≥0, are theh-vectors ofA(G), respectively of A(G¯ _F), then

(2) h(z) = X

F⊂[n]

h^F(z)(1−z)^|F|−rFz^n−|F|.

Proof. (1) can be proved exactly as in [8, Theorem 1.5].

It is known that H_A(G)(z) = _(1−z)^h(z)2n+1 (since dimA(G) = dimS+ 1 = 2n+ 1, [3]) and HA(G¯ F) = _(1−z)^hF(z)_rF+1 (since dim ¯A(GF) =rF + 1, [2]), for all F ⊂[n], henceh(z) = P

F⊂[n]

h^F(z)(1−z)^|F|−rFz^n−|F|.

Remark 4.6. By using (2) we get h_n+1 = X

F⊂[n]

(−1)^|F|−rh^F_r+1 and h_n= X

F⊂[n]

(−1)^|F|−r[h^F_r −(|F| −r)h^F_r+1],

where r = r_F = rank(L_GF). By Remark 4.3, h^F_r = h^F_r+1 = 0, for all ∅ 6=

F ⊂[n]. Hencehn+1 = h^∅₁ = 0, hn =h^∅₀ = 1 and the a-invariant of A(G) is a=−n−1. In [7, Corollary 4.4] it was proved thatA(G) is a Gorenstein ring, therefore, by [3, Corollary 4.4.6], h_i =hn−i, for all 0≤i≤n.

Corollary 4.7. Let Gbe an unmixed bipartite graph on Vn=W∪W⁰, n≥1, without isolated vertices. Then

(3) e(A(G)) = X

F⊂[n]

GFCohen-Macaulay

e( ¯A(G_F)),

where, by convention, G_∅ is considered a Cohen-Macaulay subgraph of G.

Proof. By [6, Theorem 2.2] GF is a Cohen-Macaulay bipartite graph if and only if rank(L_GF) =|F|, for all∅ 6=F ⊂[n]. Thus (3) follows immediately from (2).

We compute the Hilbert series of the vertex cover algebra of the unmixed complete bipartite graph K_n,n,n≥1.

Proposition4.8. For alln≥1H_A(Kn,n)(z) = ^1+z+···+z_(1−z)2n+1ⁿ. In particular, the multiplicity e(A(K_n,n)) + 1.

Proof. L_Kn,n = {∅,{p₁, p₂, . . . , p_n}}, therefore, by Theorem 3.1, Q_Kn,n is a principal ideal generated by b = x1. . . xnu2 −y1. . . ynu1, where u1 = u_{p1,...,pn} and u2 = u_∅. Then A(Kn,n) ' B_Kn,n/Q_Kn,n and the minimal graded free resolution of A(K_n,n) is given by the short exact sequence 0 → BKn,n(−(n+ 1))→^·b BKn,n → A(Kn,n) → 0. HenceH_A(Kn,n)(z) = ^1+z+···+z_(1−z)2n+1ⁿ. In particular, the multiplicity e(A(K_n,n)) + 1.

LetP_n ={p₁, p₂, . . . , p_n} be a poset with a partial order ≤. We denote by G(Pn) the bipartite graph on Vn=W ∪W⁰, whose edge setE(G) consists of all 2-element subsets {x_i, y_j} withp_i ≤p_j. It is said that a bipartite graph G on V_n = W ∪W⁰ comes from a poset, if there exists a finite poset P_n on {p₁, p2, . . . , pn} such that pi ≤ pj implies i ≤ j, and after relabelling of the vertices of G one hasG=G(P_n).

Corollary 4.9. Let Gbe an unmixed bipartite graph on V_n=W∪W⁰, n≥1, without isolated vertices. Then

n+ 1≤e(A(G))≤n!

n

X

l=0

1 l!.

The left equality holds if and only if G=Kn,n and the right equality holds if and only if Gcomes from an antichain with n elements.

Proof. By (3) and [1, Proposition 3.4(3)], e(A(G)) = P

F⊂[n], r=|F|

f_r^F, where r = rank(L_GF) and f_r^F is the last component of the f-vector of the order complex ∆(L_GF). Then e(A(G)) + 1 + P

F⊂[n], r=|F|≥2

f_r^F, which implies

that e(A(G))≥n+ 1. The equality holds if and only if rank(L_GF)<|F| for all F ⊂ [n] with |F| ≥ 2, which is equivalent to G = K_n,n. On the other hand, e(A(G)) = 1 + P

F⊂[n], r=|F|≥1

f_r^F. If rank(L_GF) = |F|, ∅ 6= F ⊂ [n], then L_GF is a full sublattice of a Boolean lattice on a set with |F|elements, hence f_|F^F_|≤ |F|! and e(A(G))≤1 + P

F⊂[n],|F|≥1 n

|F|

|F|!!Pⁿ

l=0 1

l!. The equality holds if and only if L_GF is a Boolean lattice on a set with|F|elements, for all

∅ 6=F ⊂[n], which is equivalent to saying thatGcomes from an antichain.

Remark 4.10. In general unmixed bipartite graphs are not uniquely de- termined, up to a graph isomorphism, by the h-vector of their corresponding vertex cover algebras. Let G3 be the bipartite graph on V3 = {x₁, x2, x3} ∪ {y₁, y₂, y₃}with the edge set{x₁, y₁},{x₂, y₂},{x₃, y₃},{x₂, y₃},{x₃, y₂}and G⁰₃ be the bipartite graph on V3 ={x₁, x2, x3} ∪ {y₁, y2, y3} that comes from the chain P₃⁰ = {p⁰₁, p⁰₂, p⁰₃} with p⁰₁ ≤ p⁰₂ ≤p⁰₃. G₃ and G⁰₃ are unmixed and they are not isomorphic, and by computation we get

H_A(G3)(z) =H_A(G⁰

3)(z) = 1 + 3z+ 3z²+z³ (1−z)⁷ .

However, unmixed complete bipartite graphs and bipartite graphs that come from chains and antichains are uniquely determined, up to a graph isomorphism, by the h-vector of their corresponding vertex cover algebras.

The statement for bipartite graphs that come from chains and antichains was proved in [8, Proposition 2.3].

Corollary4.11. LetGbe an unmixed bipartite graph on Vn=W∪W⁰, n ≥ 1, without isolated vertices. Then G = K_n,n if and only if H_A(G)(z) =

1+z+···+zⁿ (1−z)²ⁿ⁺¹ .

Proof. (“If”) By using (2) we get h₁ = h^[n]₁ + rank(L_G). Since h^[n]₁ is the component of rank 1 in the h-vector of ¯A(G), by using the formula which relates theh-vector to thef-vector of the order complex ∆(L_G), we geth^[n]₁ =

|L_G| −rank(L_G)−1,which implies thath₁=|L_G| −1. By hypothesis,h₁= 1, hence |L_G| = 2. G is an unmixed bipartite graph on Vn, therefore, L_G = {∅,{p₁, . . . , p_n}}and G=K_n,n.

(“Only if”) It follows from Proposition 4.8.

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Received 26 May 2010 Ovidius University

Faculty of Mathematics and Computer Science Bd. Mamaia 124, 900527 Constant¸a, Romania

[email protected]