Vanishing ideals over complete multipartite graphs
aa r X i v : . [ m a t h . A C ] S e p VANISHING IDEALS OVERCOMPLETE MULTIPARTITE GRAPHS
JORGE NEVES AND MARIA VAZ PINTO
Abstract.
We study the vanishing ideal of the parametrized algebraic toric set associated tothe complete multipartite graph G = K α ,...,α r over a finite field of order q . We give an explicitfamily of binomial generators for this lattice ideal, consisting of the generators of the ideal ofthe torus, (referred to as type I generators), a set of quadratic binomials corresponding to thecycles of length 4 in G and which generate the toric algebra of G (type II generators) and a set ofbinomials of degree q − G (type III generators). Using this explicitfamily of generators of the ideal, we show that its Castelnuovo–Mumford regularity is equal tomax { α ( q − , . . . , α r ( q − , ⌈ ( n − q − / ⌉} , where n = α + · · · + α r . introduction The class of vanishing ideals of parameterized algebraic toric sets over a finite field was first studiedby Renteria, Simis and Villarreal in [18]. Here we focus on the case when the set is parameterizedby the edges of a simple graph. Let K be a finite field of order q and G a simple graph with n vertices { v , . . . , v n } and nonempty edge set. Given a choice of ordering of the edges, given by abijection e : { , . . . , s } → E ( G ), and writing x e ( i ) = x j x k for every x = ( x , . . . , x n ) ∈ ( K ∗ ) n and e ( i ) = { v j , v k } ∈ E ( G ), we define the associated algebraic toric set as the subset of P s − given by:(1.1) X = (cid:8) ( x e , . . . , x e s ) ∈ P s − : x ∈ ( K ∗ ) n (cid:9) , where we abbreviate the notation e ( i ) to e i .The variety X can also be seen as the subgroup of T s − ⊂ P s − given by the image of the grouphomomorphism ( K ∗ ) n → T s − defined by x ( x e , . . . , x e s ). The vanishing ideal of X , which wedenote by I ( X ), is the ideal generated by all homogeneous forms in S = K [ t , . . . , t s ] that vanishon X . This ideal is a Cohen–Macaulay, radical, lattice ideal of codimension s − cf. [18, Theorem2.1]). One motivation for the study of these ideals lies in the fact that they combine the toric idealof the edge subring of a graph, P ( G ) ⊂ K [ t , . . . , t s ], with the arithmetic of the finite field. Thisrelation is expressed in the equality:(1.2) I ( X ) = (cid:0)(cid:2) P ( G ) + ( t q − − t q − , . . . , t q − s − t q − ) (cid:3) : ( t · · · t s ) ∞ (cid:1) Mathematics Subject Classification.
Primary 13C13; Secondary 13P25, 14G15, 14G50, 11T71, 94B05, 94B27.The first author was partially funded by CMUC, through European program COMPETE/FEDER and FCTproject PEst-C/MAT/UI0324/2011. Part of this work was developed during a research visit to to CINVESTAV ofthe IPN, M´exico, in which the first author benefited from the financial support of a research grant from SantanderTotta Bank (Portugal). The second author is a member of the Center for Mathematical Analysis, Geometry, andDynamical Systems, Departamento de Matematica, Instituto Superior Tecnico, 1049-001 Lisboa, Portugal. which (in particular) holds for any connected or bipartite G ( cf. [18, Corollary 2.11]). Recallthat P ( G ) is the kernel of the epimorphism K [ t , . . . , t s ] → K [ G ] given by t i y e i , where K [ G ] is the edge subring of G , i.e. , the subring of the polynomial ring K [ y , . . . , y n ] given by K [ G ] = K [ y e i : i = 1 , . . . , s ]. For a survey on the subject of toric ideals of edge subrings of graphswe refer the reader to [6, Chapter 5].The properties of I ( X ) (even in the general case of disconnected graphs) are reflected in G and viceversa . For one, the degree (or multiplicity) of S/I ( X ) is equal to(1.3) (cid:0) (cid:1) γ − ( q − n − m + γ − , if γ ≥ q is odd , ( q − n − m + γ − , if γ ≥ q is even , ( q − n − m − , if γ = 0 , where m is the number of connected components of G , of which exactly γ are non-bipartite ( cf . [17,Theorem 3.2]). Another invariant of interest is the index of regularity of the quotient S/I ( X ), which,since this quotient is Cohen–Macaulay of dimension 1, coincides with the Castelnuovo–Mumfordregularity. To present day knowledge, there is no single general formula expressing the regularityof S/I ( X ) in terms of the data of G . It is known that when G = C k , an even cycle of length 2 k , theregularity of S/I ( X ) is ( k − q −
2) ( cf. [17, Theorem 6.2]). In the case of an odd cycle, X coincideswith T s − ( cf. [18, Corollary 3.8]) — another way of seeing this, using (1.2), is that if G is an oddcycle then P ( G ) = (0) ( cf . [24]); accordingly I ( X ) = ( t q − − t q − , . . . , t q − s − t q − ) is a completeintersection (see also [8, Theorem 1]). In this case the regularity is ( s − q −
2) = ( n − q − n (odd) is the number of vertices (and edges) of G ( cf. [8, Lemma 1]). If G = K a,b is acomplete bipartite graph, the regularity of S/I ( X ) is given bymax { ( a − q − , ( b − q − } ( cf. [7, Corollary 5.4]). Recently, a formula for the regularity of S/I ( X ) in the case of a completegraph G = K n was given in [9]. In this case, if n > S/I ( X ) = ⌈ ( n − q − / ⌉ . Notice that the case G = K is trivial and case G = K = C was already discussed.In this work we focus on the case of G = K α ,...,α r , a complete multipartite graph with α + · · · + α r = n vertices. One of our main results, Theorem 4.3, states that in this case, if r ≥ n ≥ S/I ( X ) = max { α ( q − , . . . , α r ( q − , ⌈ ( n − q − / ⌉} . This formula generalizes (1.4); it contains the case of the complete graph by setting α = · · · = α r =1. However, as far as the proof of Theorem 4.3 is concerned, we restrict to the case when K α ,...,α r isnot a complete graph. Moreover, the methods used in this work are distinctly orthogonal to thoseused in [9]. Our main interest being the lattice ideal I ( X ), we rely on a precise description of agenerating set of binomials to prove the statement on the regularity. In Theorem 3.3, we show thata given set of binomials generates I ( X ). These binomials are classified into 3 classes: the binomials t q − i − t q − j , for every i = j , which, by (1.2), belong to I ( X ) no matter which G we take, and are DEALS OVER COMPLETE MULTIPARTITE GRAPHS 3 referred to as type I generators ; the binomials t i t j − t k t l ∈ P ( G ), for each e i e k e j e l cycle of length 4contained in G , are referred to as type II generators ; finally the type III generators , obtained fromweighted subgraphs of G , are described in full detail in the beginning of Section 3. Theorem 3.3applies without restrictions on α , . . . , α r . In particular, it yields a generating set for I ( X ) in thecase of a complete bipartite graph, which, despite the result on the regularity of S/I ( X ) in [7], wasmissing in the literature.This problem area has been attracting increasing interest. The field of binomial ideals has beenquite explored and its general theory can be found in Eisenbud and Sturmfels article [4]. In ourpresent setting, these binomial ideals have a remarkable application to coding theory. Associatingto X an evaluation code, one can relate two of its basic parameters (the length and the dimension)to I ( X ) by a straightforward application of the Hilbert function ( cf. [11, 12]); moreover a set ofgenerators of I ( X ) can make way to computing the Hamming distance of the code ( cf. [19]). Therehas been substantial recent research exploring the relation to coding theory ( cf . [17, 18, 19, 20])and also focusing on the vanishing ideal of parameterized algebraic toric sets ( cf. [15, 16]).Let us describe the structure of this paper. In Section 2 we establish the definitions and notationsused in the article. In that section, Lemma 2.3 provides a useful characterization of a binomial in I ( X ) by a condition on the associated weighted subgraph of G . In Section 3, we describe 3 familiesof binomials and prove that they form a generating set for I ( X ) — Theorem 3.3. In Section 4, weprove Theorem 4.3, that states that under the assumption that r ≥ n = α + · · · + α r ≥ S/I ( X ) is given by the integer d of formula (1.5). We show this by: i) exhibiting amonomial in K [ t , . . . , t s ] of degree d and showing that that monomial does not belong to I ( X )+( t ),where t ∈ K [ t , . . . , t s ] is a variable; ii) and by showing that every monomial in K [ t , . . . , t s ] ofdegree d + 1 is in I ( X ) + ( t ).For any additional information in the theory of monomial ideals and Hilbert functions, we refer to[21, 23], and for graph theory we refer to [1]. Acknowledgement.
The authors thank Rafael Villarreal for many helpful discussions.2.
Preliminaries
Let K be a finite field of order q . Throughout, G = K α ,...,α r will denote the complete multipartitegraph. More precisely, a graph with vertex set V G = { v , . . . , v n } endowed with a partition V G = P ⊔ · · · ⊔ P r , satisfying P i = α i , for i = 1 , . . . , r , and α + · · · + α r = n , such that { v k , v l } is anedge of G if and only if v k ∈ P i and v l ∈ P j with i = j . An important case to consider is that whenall α i = 1, i.e. , when G is the complete graph on r vertices. Fix an ordering of the set of edges ofthe graph, E ( G ), given by e : { , . . . , s } → E ( G ), where s = (( P ri =1 α i ) − P ri =1 α i ) /
2. We write e j for e ( j ). Let S = K [ t , . . . , t s ] be a polynomial ring on s = E ( G ) variables. We fix a bijectionbetween the set of variables of S and E ( G ), given by the map t i e i . To ease notation, we alsouse t { v k ,v l } for the variable corresponding to the edge { v k , v l } . Let X ⊂ P s − be the algebraic toricset associated to G , as defined in (1.1), and denote by I ( X ) ⊂ S its vanishing ideal. JORGE NEVES AND MARIA VAZ PINTO
The identification of a monomial (or a binomial) of S with a weighted subgraph of G will play animportant role in Sections 3 and 4. If a = ( a , . . . , a s ) ∈ N s we denote by t a the monomial of S given by t a · · · t a s s . Definition 2.1.
Given g = t a ∈ S , where a ∈ N s , the weighted subgraph associated to t a , denotedby H g , is the subgraph of G with the same vertex set, with the set of edges corresponding to thevariables dividing t a and with weight function given by a , i.e. , such that the weight of the edge e i is a i for each i ∈ supp a . The weighted degree of a vertex v is the sum of the weights of all edgesincident to v in H g .Let us denote the weight, a i , of an edge e i in H g by wt H g ( e i ) = a i . We denote the weighted degreeof v by wt H g ( v ). This number is zero if no edge in H g is incident to v . Definition 2.2.
Given a binomial, f = t a − t b ∈ S , with supp a ∩ supp b = ∅ , the weighted subgraphassociated to f , which we denote by H f , is defined by H t a ∪ H t b .The edges of H f may be colored with two colors using the partition E ( H t a ) ⊔ E ( H t b ). We willuse a solid line for edges corresponding to variables dividing t a and a dotted line for the edgescorresponding to variables dividing t b . We refer to the former as black edges and to the latter asdotted edges. Although one can also define the notion of weighted degree of a vertex in this case,we will only use it for monomials. The usual notion of degree of a vertex, i.e. , the number of edgesincident to it, disregarding weights and coloring of the edges, will be used. Lemma 2.3.
Let G be any graph and X its associated algebraic toric set. Let f = t a − t b be abinomial (not necessarily homogeneous). Then, f vanishes on X if and only if for all v ∈ V G , wt H ta ( v ) ≡ wt H tb ( v ) (mod q − .Proof. Suppose that f = t a − t b vanishes on X and let v = v l ∈ V G , for l ∈ { , . . . , n } . Let x = ( . . . , x l x l ′ , . . . ) ∈ X be given by x l = u , where u is a generator of K ∗ , and x l = 1, for any l = l . Then f ( x ) = 0 = ⇒ u wt H ta ( v l ) − u wt H tb ( v l ) = 0 ⇐⇒ wt H ta ( v l ) ≡ wt H tb ( v l ) (mod q − . Conversely, let x = ( . . . , x l x l ′ , . . . ) ∈ X be any point of X and assume that wt H ta ( v ) ≡ wt H tb ( v )(mod q − v ∈ V G . Then f ( x ) = n Y l =1 x wt H ta ( v l ) l − n Y l =1 x wt H tb ( v l ) l = 0 . (cid:3) Let I ⊂ S = K [ t , . . . , t s ] be (any) homogeneous ideal and let H S/I ( d ) = dim K S d /I d , for every d ≥
0, be the Hilbert function of the graded ring
S/I . There exists a polynomial h S/I ( t ) in Z [ t ]of degree k −
1, where k = dim S/I , such that H S/I ( d ) = h S/I ( d ) for d ≫
0. The degree or multiplicity of S/I is, by definition, the leading coefficient of h S/I ( t ) multiplied by ( k − index of regularity of S/I is the least integer ℓ ≥ H S/I ( d ) = h S/I ( d ) for d ≥ ℓ . Letting DEALS OVER COMPLETE MULTIPARTITE GRAPHS 5 b ij = dim K Tor i ( K, S/I ) j be the graded Betti numbers of S/I , the
Castelnuovo–Mumford regularity of S/I is, by definition, max { j − i | b ij = 0 } . If I is Cohen–Macaulay and dim S/I = 1, thenthe index of regularity of
S/I coincides with the Castelnuovo–Mumford regularity of
S/I (cf. [23,Corollary 2.5.14 and Proposition 4.2.3 ]). As this is the context of this work, with I = I ( X ) ( cf. [18,Theorem 2.2]), we will denote both by reg( S/I ( X )).Hence, in our case, the Hilbert function of the ring S/I ( X ) is constant for d ≥ reg( S/I ( X )). Theconstant value at which it stabilizes is | X | , and since the Hilbert function of S/I ( X ) is strictlyincreasing in the range 0 ≤ d ≤ reg( S/I ( X )) (cf. [3], [5]), the regularity of S/I ( X ) is equal to thefirst d for which it attains the value | X | . Note that | X | is equal to ( q − n − by the formula (1.3)— recall that n = V G . 3. A generating set for I ( X )In this section we describe a generating set for the vanishing ideal I ( X ) of the algebraic toric setassociated to a complete multipartite graph G = K α ,...,α r . We will distinguish 3 types of generators.The type I generators are of the form t q − i − t q − j , for 1 ≤ i, j ≤ s . The type II generators are in1-to-1 correspondence with the cycles of length 4, { e i , e j , e k , e l } , in G ( cf. see Figure 1). Each cycleyields the generator: t i t k − t j t l . • v i • v i • v i • v i ❅❅❅❅ e i ··········· e k ❅❅❅ e j ··········· e l Figure 1.
A 4-cycle yieldingtype II generator. • v l n ′ +1 ····················· e j ···················· e j ··· ···· ····· ····· ··· e j ··· ···· ····· ····· ····· ····· e j n ′ ···• v l • v l • v l • v l n ′ ❤❤❤❤❤❤❤❤ e i ✭✭✭✭✭✭✭ e i ✱✱✱✱✱ e i ✆✆✆✆✆✆✆✆✆ e i n ′ • v l Figure 2.
A subgraph yield-ing a type
III generator.A type III generator is specified by the choice of n ′ ≥ v l , . . . , v l n ′ ∈ V G , plus 2additional vertices v l and v l n ′ +1 such that the edges { v l , v l k } and { v l n ′ +1 , v l k } exist, for all 1 ≤ k ≤ n ′ and, furthermore, by the choice of positive integers 1 ≤ d k ≤ q − d + · · · + d n ′ = q − Q n ′ k =1 t d k i k − Q n ′ k =1 t d k j k , where, for each k ∈ { , . . . , n ′ } , i k and j k are such that e i k = { v l , v l k } and e j k = { v l n ′ +1 , v l k } . The associated weighted subgraph of G is depicted in Figure 2. By a straightforward application of Lemma 2.3, it is clear that thehomogeneous binomials of the 3 types listed earlier belong to I ( X ).We will need the following lemma. JORGE NEVES AND MARIA VAZ PINTO
Lemma 3.1.
Let v l , v l , . . . , v l n ′ , v l n ′ +1 ∈ V G , be such that n ′ ≥ and the edges { v l , v l k } and { v l n ′ +1 , v l k } exist in G , for all ≤ k ≤ n ′ . Let ≤ d k ≤ q − be such that d + · · · + d n ′ = α ( q − ,where α is a positive integer. Consider f = Q n ′ k =1 t d k i k − Q n ′ k =1 t d k j k , where, for each k ∈ { , . . . , n ′ } , i k and j k are such that e i k = { v l , v l k } and e j k = { v l n ′ +1 , v l k } . Then f belongs to the ideal generatedby the type III binomials.Proof. We proceed by induction on α . If α = 1, f is exactly a type III generator. Assume α ≥ m ∈ N such that 2 ≤ m ≤ n ′ and, for each k = 1 , . . . , m , choose d ′ k ∈ N such that1 ≤ d ′ k ≤ d k ≤ q − d ′ + · · · + d ′ m = q −
1. Then Q mk =1 t d ′ k i k − Q mk =1 t d ′ k j k is a type III generator.Write f = t a Q mk =1 t d ′ k i k − t b Q mk =1 t d ′ k j k , for appropriate a, b ∈ N s . Then: f = Q n ′ k =1 t d k i k − Q n ′ k =1 t d k j k = (cid:16)Q mk =1 t d ′ k i k − Q mk =1 t d ′ k j k (cid:17) t a + ( t a − t b ) Q mk =1 t d ′ k j k . By induction, t a − t b is in the ideal generated by the type III binomials, hence so is f . (cid:3) Lemma 3.2.
Let f = t a − t b be a (homogeneous) binomial in I ( X ) . If H f contains one of thesubgraphs depicted in Figure 3, with v = v , and either there is an edge in G through v and v or one of v , v is dichromatic, then there exists j ∈ { , . . . , s } and a (homogeneous) binomial g ∈ K [ t , . . . , t s ] such that f − t j g belongs to the ideal of K [ t , . . . , t s ] generated by the binomials oftype I,II and III. • v e i ❏❏❏❏ • v ·················· e j • v • ✡✡✡✡ v e k • v e j v ··········• e i • v •·········· v e l Figure 3.
Two special dichromatic edge arrangements
Proof.
Let J be the ideal of K [ t , . . . , t s ] generated by the binomials of type I, II and III.Case 1. Suppose, without loss of generality, that it is the case of the graph on the left of Figure 3and that there exists an edge in G through v and v . Denote this edge by e l . Then t i t k − t j t l is a type II generator. Let f = t a − t b = t i t k t a ′ − t j t b ′ , for appropriate a ′ , b ′ ∈ N s and consider g = t l t a ′ − t b ′ . Then, f − t j g = t i t k t a ′ − t j t b ′ − t j ( t l t a ′ − t b ′ ) = ( t i t k − t j t l ) t a ′ ∈ J. Case 2. Assume now that v and v have no edge in G between them, (in other words, that theybelong to the same part of the partition of V G ) and that v is dichromatic. Let e l be a dotted edgeincident to this vertex. Let v be its other endpoint. If there exists an edge in G between v and v , we reduce to Case 1 (graph on the right of Figure 3). Assume v and v belong to the samepart of the partition of V G . Observe that v and v must be in distinct parts of the partition of V G (otherwise v and v would be in that same part, but we are assuming that there is an edge e k through v and v ). Therefore there is an edge in G , denote it by e λ , through v and v . By thesame type of reasoning, we deduce that there is an edge in G , denote it by e µ , through v and v . DEALS OVER COMPLETE MULTIPARTITE GRAPHS 7 If v = v , then e λ = e l , which is a dotted edge in H f . Since v and v are in different parts of thepartition of V G , we are reduced (with e i , e l and e k ) to Case 1 (graph on the left of Figure 3). So wemay assume v = v . Write f = t i t k t a ′ − t b and consider f ′ = t λ t µ t a ′ − t b . Notice that e λ is a blackedge of the subgraph H f ′ , and since t l and t j divide t b , e l and e j are dotted edges of the subgraph H f ′ . Observe also that v and v are not in the same part of the partition of V G (otherwise v and v would be in that same part, but we are assuming that there is an edge e j through v and v ).Then there must be an edge in G through v and v . Therefore, H f ′ contains a subgraph with edges e l , e λ and e j , satisfying the assumption of Case 1 (graph on the right of Figure 3). Thus, thereexists g ∈ K [ t , . . . , t s ] such that f ′ − t λ g is a multiple of a type II generator. To complete the proofit suffices to observe that f = f ′ + ( t i t k − t λ t µ ) t a ′ and that t i t k − t λ t µ is a type II generator. (cid:3) Theorem 3.3.
The binomials of type I, II and III generate I ( X ) .Proof. Denote by J the ideal of K [ t , . . . , t s ] generated by the binomials of type I, II and III. Clearly J ⊆ I ( X ). By [17, Theorem 4.5] there exists a set of generators of I ( X ) consisting of the generatorsof type I, plus a finite set of homogeneous binomials t a − t b with supp( a ) ∩ supp( b ) = ∅ and such thatthe degree of t a − t b in each variable is ≤ q −
2. Hence it will suffice to show that any homogeneousbinomial f = t a − t b ∈ I ( X ) satisfying the latter conditions belongs to the ideal generated by theelements in the 3 classes described, or, equivalently, is congruent to 0 modulo J . Let f = t a − t b be such a binomial and, as defined above, let H f be the induced subgraph of G . We will argue byinduction on V + H f + deg( f ), where V + H f denotes the subset of V H f = V G of vertices with positivedegree.By Lemma 2.3, no vertex of H f has (standard) degree equal to 1. Hence V + H f ≥ V + H f = 3 then H f reduces to a triangle. This situation is impossible, since f is homogeneousand the condition given in the statement of Lemma 2.3 must be satisfied. The base case is thus V + H f = 4 and deg( f ) = 2. H f must reduce to a square, and using Lemma 2.3 we deduce that f is a type II generator if q > q = 3.Suppose that V + H f +deg( f ) ≥
7. If all vertices of H f are dichromatic then we can find a subgraph of H f satisfying the assumptions of Lemma 3.2. Then there exists j ∈ { , . . . , s } such that f − t j g ∈ J ,for some homogeneous binomial g ∈ K [ t , . . . , t s ]. Since J ⊂ I ( X ), f ∈ I ( X ) and t j does not vanishon any point of X , we deduce that g ∈ I ( X ). It is clear that deg( g ) < deg( f ). We may assumethat g = t a ′ − t b ′ with supp( a ′ ) ∩ supp( b ′ ) = ∅ (dividing through by an appropriate monomial,if necessary). We may also assume that 0 ≤ a ′ i , b ′ i ≤ q − t a ′ l l divides t a ′ and a ′ l ≥ q −
1; let t k divide t b ′ ; then g = t q − l t a ′′ − t k t b ′′ + t q − k t a ′′ − t q − k t a ′′ = ( t q − l − t q − k ) t a ′′ + t k h , where h = t a ′ l − ( q − l t c − t d , for appropriate a ′′ , b ′′ , c, d ∈ N s ; since t q − l − t q − k is a type I generator, g ∈ I ( X )if and only if h ∈ I ( X ). Repeating the argument, we may indeed assume that 0 ≤ a ′ i , b ′ i ≤ q − g ≡ J , which implies that f ≡ J . JORGE NEVES AND MARIA VAZ PINTO
Let now v l be a monochromatic vertex of H f . Assume, without loss of generality, that all of itsincident edges are black. Denote by v l , . . . , v l n ′ ( n ′ ≥
2) their endpoints and let e i k = { v l , v l k } ,for k = 1 , . . . , n ′ .Special condition: Suppose there exists v l n ′ +1 ∈ V + H f with v l n ′ +1 = v l and such that, for all k = 1 , . . . , n ′ , { v l n ′ +1 , v l k } ∈ E G . Denote these edges by e j k . By Lemma 2.3, a i + · · · + a i n ′ ≡ q −
1) and therefore, by Lemma 3.1, Q n ′ k =1 t a ik i k − Q n ′ k =1 t a ik j k ∈ J . Writing t a = t a ′ Q n ′ k =1 t a ik i k ,for suitable a ′ ∈ N s , f = t a ′ Q n ′ k =1 t a ik i k − t b ≡ t a ′ Q n ′ k =1 t a ik j k − t b mod J. Write g = f − t a ′ ( Q n ′ k =1 t a ik i k − Q n ′ k =1 t a ik j k ) = t a ′ Q n ′ k =1 t a ik j k − t b = t c − t b , for appropriate c ∈ N s . Then g ∈ I ( X ), the graph induced by g has one vertex of positive weighted degree fewer than the graph H f , and g has degree equal to deg f . As above, by using a type I generator and dividing throughby an appropriate monomial (in which case the resulting monomial would have degree strictly lessthat deg f ), we may also assume that no variable in g occurs to a higher power than q − c ) ∩ supp( b ) = ∅ . By induction, we deduce that f ≡ g ≡ J .Denote now by P µ , P µ , . . . , P µ m the parts of the partition of V G that have nonempty intersectionwith V + H f , with v l ∈ P µ . If P µ ∩ V + H f ) ≥
2, we are in the special condition case, and therefore f ≡ J . We may assume that P µ ∩ V + H f = { v l } . We may also assume that P µ k ∩ (cid:8) v l , . . . , v l n ′ (cid:9) = ∅ , for all k = 1 , . . . , m , for otherwise, if for some k ∈ { , . . . , m } , this intersection is empty, choosing v l n ′ +1 ∈ P µ k ∩ V H + f , we are again in the special condition case. Another observation is the following:if there exists k ∈ { , . . . , m } such that P µ k ∩ V H + f ) ≥ P µ k ∩ V H + f ismonochromatic, then, using the same argument we used for P µ ∩ V H + f , we conclude that f ≡ J . We may therefore assume that for all k ∈ { , . . . , m } such that P µ k ∩ V H + f ) ≥
2, thevertices of P µ k ∩ V H + f are all dichromatic.Applying Lemma 2.3 for each vertex in V + H f \ { v l } and summing all the congruences obtained,(3.1) P n ′ k =1 a i k + 2 P i ∗ a i ∗ = 2 P i ∗ b i ∗ + δ ( q − , where i ∗ varies on the subset of { , . . . , s } corresponding to edges of H f except for e i , . . . , e i n ′ , and δ ∈ Z is given by δ = P v ∈ V + H f \ { v l } δ v , where δ v ∈ Z yields the congruence in of Lemma 2.3 for thevertex v . Since f is homogeneous,(3.2) P n ′ k =1 a i k + P i ∗ a i ∗ = P i ∗ b i ∗ . Together, (3.1) and (3.2) imply that P n ′ k =1 a i k + δ ( q −
1) = 0. We deduce that δ <
0. This meansthat for some vertex v λ ∈ V + H f ∩ (cid:0) P µ ∪ · · · ∪ P µ m (cid:1) , δ λ <
0, and, in particular, the sum of theweights of dotted edges incident to v λ is ≥ q −
1. Denote by e ν , . . . , e ν ˜ n the dotted edges incidentto v λ and by v λ , . . . , v λ ˜ n their endpoints, so that e ν k = { v λ , v λ k } and P ˜ nk =1 b ν k ≥ q −
1. Noticethat, necessarily ˜ n ≥
2. We claim that v λ must be dichromatic. This is clear if v λ coincides withone of v l , . . . , v l n ′ . Otherwise, since v λ belongs to some P µ k and P µ k ∩ (cid:8) v l , . . . , v l n ′ (cid:9) = ∅ , we get P µ k ∩ V + H f ) ≥ v λ is not monochromatic. The same argument can be used to show DEALS OVER COMPLETE MULTIPARTITE GRAPHS 9 that v λ , . . . , v λ ˜ n are all dichromatic. For each of the vertices v λ , v λ , . . . , v λ ˜ n choose a black edgeincident to it, denote it by e γ i , for i = 0 , . . . , ˜ n , and let v β i be its endpoint.Suppose there exists r ∈ { , . . . , ˜ n } such that v β = v β r . Either there is an edge in G through v β and v β r , or v β and v β r are in the same part of the partition of V G . Then there exists k ∈ { , . . . , m } such that v β , v β r ∈ P µ k ∩ V + H f , and since P µ k ∩ V + H f ) ≥ v β and v β r are dichromatic. In anycase, we may use Lemma 3.2 and argue as previously to show that f ≡ J . Suppose then thatfor all r ∈ { , . . . , ˜ n } , the endpoints v β r all coincide. As in the proof of Lemma 3.1, choose m ′ ∈ N such that 2 ≤ m ′ ≤ ˜ n and, for each k = 1 , . . . , m ′ , choose b ′ ν k ∈ N such that 1 ≤ b ′ ν k ≤ b ν k ≤ q − b ′ ν + · · · + b ′ ν m ′ = q −
1. Then Q m ′ k =1 t b ′ νk γ k − Q m ′ k =1 t b ′ νk ν k is a type III generator, and therefore belongs to J . Writing f = t a − t b = t a ′ Q m ′ k =1 t γ k − t b ′ Q m ′ k =1 t b ′ νk ν k , for appropriate a ′ , b ′ ∈ N s , we deduce that f ≡ t a ′ Q m ′ k =1 t γ k − t b ′ Q m ′ k =1 t b ′ νk γ k = (cid:0) t a ′ − t b ′ Q m ′ k =1 t b ′ νk − γ k (cid:1) Q m ′ k =1 t γ k mod J. The homogeneous binomial g = t a ′ − t b ′ Q m ′ k =1 t b ′ νk − γ k = t a ′ − t b ′′ , which by the above belongs to I ( X ), is such that deg( g ) = deg( f ) − m ′ < deg( f ). We may assume supp( a ′ ) ∩ supp( b ′′ ) = ∅ . Byinduction, g ≡ J , which implies that f ≡ J and completes the proof. (cid:3) Regularity of
S/I ( X )Let G = K α ,...,α r be a complete multipartite graph with r ≥
3. We assume G does not coincidewith K , , , the complete graph on 3 vertices. In that case, the associated toric set X coincideswith the ambient torus, T ⊂ P and by cf. [8, Lemma 1], the regularity of S/I ( X ) is 2( q − S/I ( X ) = max { α ( q − , α ( q − , . . . , α r ( q − , ⌈ ( n − q − / ⌉} , where, n = V G = α + · · · + α r ≥
4. If, without loss in generality, we assume that α ≥ α ≥· · · ≥ α r , then this formula takes on the simpler form:reg S/I ( X ) = max { α ( q − , ⌈ ( n − q − / ⌉} . The case of the complete graph, i.e. , when α = · · · = α r = 1, is treated in [9].In the proof of this result we will need to show that reg S/I ( X ) ≥ ⌈ ( n − q − / ⌉ . Thisinequality was shown to hold for any graph for which X = ( q − n − in [10]. Indeed [10,Corollary 3.13] implies that if X is the algebraic toric set associated to a k -uniform clutter on n vertices then reg S/I ( X ) ≥ (cid:24) ( X )( n − q − k ( q − n − (cid:25) . In the proof of Theorem 4.3 we argue on an Artinian reduction of
S/I ( X ). More precisely, if f ∈ S is a regular element on S/I ( X ) of degree 1, the short exact sequence0 → S/I ( X )[ − f −→ S/I ( X ) → S/ ( I ( X ) + ( f )) → S/I ( X ) = reg S/ ( I ( X ) + ( f )) −
1. Accordingly, showing that reg
S/I ( X ) = d amounts toproving that every monomial of degree d + 1 belongs to I ( X ) + ( f ) and that there exists a monomialof degree d that does not. For a detailed explanation of this fact, see [17, Theorem 6.2]. We beginwith two lemmas. Lemma 4.1.
Let t a ∈ S be a monomial and let H t a be the associated weighted subgraph of G .Let v ∈ V G and { e i k = { v , v i k } : k = 1 , . . . , n ′ } , n ′ ≥ , be a subset of the edges of H t a incidentto v , such that P n ′ k =1 wt H ta ( e i k ) ≥ α ( q − , for some positive integer α . Let w ∈ V G \ { v } besuch that { w , v i k } ∈ E ( G ) , for all k = 1 , . . . , n ′ . Then there exists a monomial t b ∈ S such that t a − t b ∈ I ( X ) , wt H tb ( v ) = wt H ta ( v ) − α ( q − , wt H tb ( w ) ≥ P n ′ k =1 wt H tb { w , v i k } ≥ α ( q − and, for all v ∈ V G \ { v , w } , wt H ta ( v ) = wt H tb ( v ) .Proof. We may write t a = (cid:0)Q n ′ k =1 t d k i k (cid:1) t a ′ , where P n ′ k =1 d k = α ( q − a ′ ∈ N s . Denote { w , v i k } by e j , . . . , e j n ′ . Consider t b = (cid:0)Q n ′ k =1 t d k j k (cid:1) t a ′ . Then, by Lemma 3.1, t a − t b = (cid:0)Q n ′ k =1 t d k i k − Q n ′ k =1 t d k j k (cid:1) t a ′ ∈ I ( X ) . Additionally, wt H tb ( v ) = wt H ta ′ ( v ) = wt H ta ( v ) − α ( q −
1) andwt H tb ( w ) ≥ P n ′ k =1 wt H tb { w , v i k } ≥ P n ′ k =1 d k = α ( q − ≥ q − . Finally, if v is not an endpoint of e i , . . . , e i n ′ , e j , . . . , e j n ′ then wt H ta ( v ) = wt H ta ′ ( v ) = wt H tb ( v ),and otherwise, wt H ta ( v i k ) = d k + wt H ta ′ ( v i k ) = wt H tb ( v i k ), for all k = 1 , . . . , n ′ . (cid:3) Lemma 4.2.
Let t a ∈ S be a monomial and let H t a be the associated weighted subgraph of G .Given i, j ∈ { , . . . , r } such that i = j , let ∆ aj be the total weight of edges between the vertices of V G \ P j , let v i ∈ P i be a vertex, and let δ aij ≤ ∆ aj be the total weight of edges between v i and thevertices of V G \ P j . Suppose that there exists an edge { w , w } ∈ E ( H t a ) with w , w P j and w , w = v i . Then, there exists a monomial t b ∈ S such that t a − t b ∈ I ( X ) , wt H ta ( v ) = wt H tb ( v ) ,for all v ∈ V G , and such that the total weight of edges between v i and the vertices of V G \ P j , δ bij , isequal to min (cid:8) wt H ta ( v i ) , ∆ aj (cid:9) .Proof. Notice that wt H ta ( v i ) ≥ δ aij . We argue by induction on wt H ta ( v i ) − δ aij . If wt H ta ( v i ) = δ aij ,we choose t b = t a and there is nothing to prove. Suppose that wt H ta ( v i ) > δ aij . Then there existsan edge { v i , w } ∈ E ( H t a ) with w ∈ P j . Since { w , w } is an edge, one of its vertices does notbelong to P i . Assume that w P i . Then t { v i ,w } t { w ,w } − t { v i ,w } t { w ,w } is a generator of I ( X ) oftype II. Writing t a = t { v i ,w } t { w ,w } t a ′ , for suitable a ′ ∈ N s , and t c = t { v i ,w } t { w ,w } t a ′ , it is clearthat t a − t c ∈ I ( X ); moreover the numbers ∆ cj and wt H tc ( v i ) are the same as they were for H t a ,however δ cij = δ aij + 1. By induction, there exists a monomial t b ∈ S such that t c − t b ∈ I ( X ), hence t a − t b ∈ I ( X ), and such that δ bij = min (cid:8) wt H tc ( v i ) , ∆ cj (cid:9) = min (cid:8) wt H ta ( v i ) , ∆ aj (cid:9) . Also by induction, DEALS OVER COMPLETE MULTIPARTITE GRAPHS 11 wt H tc ( v ) = wt H tb ( v ), for all v ∈ V G . On the level of the graph, the induction step merely takes twoedges and swaps a pair of their endpoints. This does not change the weighted degree; thereforewt H tc ( v ) = wt H ta ( v ), for all v ∈ V G , and hence, wt H ta ( v ) = wt H tb ( v ), for all v ∈ V G . (cid:3) Theorem 4.3.
Let X be the algebraic toric set associated to an r -partite complete graph G = K α ,...,α r ,with r ≥ and n = α + · · · + α r ≥ . Then reg S/I ( X ) = max { α ( q − , α ( q − , . . . , α r ( q − , ⌈ ( n − q − / ⌉} . Proof.
Fix vertices v i ∈ P i ⊂ V G . Without loss in generality, let t be the variable t { v ,v } . Weassume that α ≥ α ≥ · · · ≥ α r . If α = 1 then G is the complete graph of n vertices andreg S/I ( X ) = ⌈ ( n − q − / ⌉ , as was proved in [9]. From now on we assume that α ≥
2. By[10, Corollary 3.13], reg
S/I ( X ) ≥ ⌈ ( n − q − / ⌉ . Let us show that reg S/I ( X ) ≥ α ( q − α ( q −
2) given by t a = t q − { v ,v } Y w ∈ P \{ v } t q − { w,v } , and consider H t a , the weighted subgraph of G induced by t a , which is depicted in Figure 4. • v • q − • ✏✏✏✏✏✏ q − ... • (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) q − ✛✚ ✘✙ P • ... •• v q − ✛✚ ✘✙ P v •• ... • ✛✚ ✘✙ P v •• ... • ✛✚ ✘✙ P . . . v r •• ... • ✛✚ ✘✙ P r Figure 4.
The weighted subgraph associated to t a .Let us show that t a I ( X ) + ( t ). Let { f , . . . , f m } be a Gr¨obner basis of I ( X ). Since, byTheorem 3.3, I ( X ) is generated by binomials, we may assume that f i is a binomial, for every i = 1 , . . . , m ( cf. [4, Proposition 1.1 (a)], a straighforward consequence of Buchberger’s algoritm).Since a variable t i never vanishes on a point of X , we may also assume that each f i = t a − t b with supp a ∩ supp b = ∅ . Choose a monomial order in which t is the least variable. Then, since f , . . . , f m , t generate I ( X ) + ( t ), and t does not divide the leading term of any f , . . . , f m , weconclude, by Buchberger’s algorithm ( cf. [23, Theorem 2.4.15] and [2]), that { t , f , . . . , f m } is aGr¨obner basis of I ( X ) + ( t ).Suppose that t a ∈ I ( X ) + ( t ). Then the division algorithm of t a by t , f , . . . , f m produces zeroremainder. Since t does not divide t a , there exists i such that Lt( f i ) divides t a . Write Lt( f i ) = t b and, without loss in generality, f i = t b − t c , for some b, c ∈ N s . Then there exists a ′ ∈ N s such that t a = t b t a ′ . Let t a = t a − t a ′ f i = t a ′ t c be the first partial reduction of t a modulo t , f , . . . , f m . Denote by t a = t a , t a , . . . , t a p the fullsequence of partial reductions of t a modulo t , f , . . . , f m . Since division of a monomial by one of f i always yields a nonzero monomial, we conclude that there must exist p > t a p isdivisible by t (and, accordingly, t a p +1 = 0). Write t a p = t t w , for some w ∈ N s . From the fact that t a i +1 − t a i ∈ I ( X ), for all i = 0 , . . . , p −
1, we deduce that t a − t t w ∈ I ( X ) . Consider H t a and H t t w the weighted subgraphs of G induced by t a and t t w , respectively. Bythe definition of t a (see also Figure 4), wt H ta ( v ) = 0 and, if v ∈ P \ { v } , wt H ta ( v ) = q − H t tw ( v ) ≥ q − v ∈ P \ { v } , thatwt H t tw ( v ) ≥ q −
2. We get: α ( q −
2) = deg( t a ) = deg( t t w ) ≥ P v ∈ P wt H t tw ( v ) ≥ α ( q −
2) + 1 , which is a contradiction. We just showed that t a I ( X ) + ( t ), hence reg S/I ( X ) ≥ α ( q − d = max { α ( q − , ⌈ ( n − q − / ⌉} . Let us now show that any monomial t a ∈ S of degree d +1 belongs to I ( X )+( t ), or, equivalently, that there exists t b ∈ ( t ) such that t a ≡ t b mod I ( X ).As earlier, we will argue on H t a , the weighted subgraph of G induced by t a .Let w , w ∈ V G such that { w , w } ∈ E ( H t a ). Write t i = t { w ,w } and t a = t bi t a ′ , where b ≥ a ′ ∈ N s . Suppose b ≥ q −
1. If t i = t , then t a ∈ ( t ) + I ( X ). If t i = t , then g = t q − i − t q − isa type I generator of I ( X ). Writing t a = t q − i t a ′′ , for some a ′′ ∈ N s , we have t a = t q − t a ′′ + gt a ′′ ∈ ( t ) + I ( X ). Therefore, we may assume that if t bi divides t a , then b ≤ q −
2. This implies we mayalso assume that if v ∈ V G is such that wt H ta ( v ) ≥ q −
1, then there exist at least two edges in H t a that have v as an endpoint.We start by deriving a basic inequality. Suppose that v ∈ P i \ { v , . . . , v r } has wt H ta ( v ) ≥ q − w = v i and α a positive integer such that ( α + 1)( q − > wt H ta ( v ) ≥ α ( q − t b ∈ S such that t a ≡ t b mod I ( X ) and wt H tb ( v ) = wt H ta ( v ) − α ( q − ≤ q −
2. Accordingly, we may assume that(4.1) for all v
6∈ { v , . . . , v r } , wt H ta ( v ) ≤ q − . Hence, P v ∈ V G wt H ta ( v ) ≤ P rj =1 wt H ta ( v j ) + ( n − r )( q − . Since P v ∈ V G wt H ta ( v ) = 2( d + 1) ≥ (cid:6) n − ( q − (cid:7) + 2 ≥ ( n − q −
2) + 2, we get(4.2) P rj =1 wt H ta ( v j ) ≥ ( r − q −
2) + 2 . Case 1.
Suppose that wt H ta ( v ) , wt H ta ( v ) > Subcase 1.1.
Suppose v and v have edges in H t a connecting them to 2 vertices in distinct P i , say { v , w } , { v , w } , with w ∈ P i and w ∈ P j , and i = j . If w = v or w = v , then t divides t a and we are done. If w = v and w = v , then since { w , w } belongs to E ( G ), we have that t { v ,w } t { v ,w } − t t { w ,w } is a type II generator of I ( X ). As, by assumption, t { v ,w } t { v ,w } divides DEALS OVER COMPLETE MULTIPARTITE GRAPHS 13 t a , we may write t a = t { v ,w } t { v ,w } t a ′ , for a suitable a ′ ∈ N s . Then, t a ≡ t t { w ,w } t a ′ mod I ( X ),as required. Subcase 1.2.
Suppose there exists i ∈ { , . . . , r } such that all edges in H t a from v and v haveendpoints in P i . Clearly i ∈ { , . . . , r } . Assume, additionally, that there exists and edge { w , w } ∈ E ( H t a ), such that w , w P i . Then, necessarily w , w = v and w , w = v and at least oneof { v , w } or { v , w } belongs to E ( G ). Assume this is the case with { v , w } . Let z ∈ P i be anendpoint of an edge in H t a incident to v . Then t { v ,z } t { w ,w } − t { v ,w } t { z,w } is a type II generatorof I ( X ) and writing t a = t { v ,z } t { w ,w } t a ′ , for a suitable a ′ ∈ N s , we get t a ≡ t { v ,w } t { z,w } t a ′ mod I ( X ). Denote t e = t { v ,w } t { z,w } t a ′ . Then t e satisfies the assumptions of Subcase 1.1 as w P i , { v , w } ∈ E ( H t e ), and all the edges in E ( H t e ) starting at v have the other endpoint in P i . Subcase 1.3.
Suppose there exists i ∈ { , . . . , r } such that all edges in H t a from v and v haveendpoints in P i . Suppose in addition that all other edges in H t a are also incident to vertices of P i .Using the assumption on deg( t a ) and (4.1), we get(4.3) P v ∈ P i wt H ta ( v ) = d + 1 ≥ α ( q −
2) + 1 ≥ q −
2) + 1 = ( q −
2) + ( q − H ta ( v i ) = d + 1 − P v ∈ P i \{ v i } wt H ta ( v ) ≥ α i ( q −
2) + 1 − ( α i − q −
2) = q − . The first inequality implies that either, ∆ a , the total weight of the edges in H t a between the verticesof V G \ P is ≥ q −
1, or ∆ a , the total weight of edges in H t a between the vertices of V G \ P is ≥ q −
1. Assume, without loss of generality that the latter is true.Let { w , w } ∈ E ( H t a ) be an edge with w , w P and assume w , w = v i . Then, by Lemma 4.2(with j = 2), there exists t b ∈ S such that t a ≡ t b mod I ( X ), wt H tb ( v ) = wt H ta ( v ) >
0, and suchthat the total weight of the edges of H t b between v i and V G \ P , δ bi , is equal to min (cid:8) wt H ta ( v i ) , ∆ a (cid:9) .From the second inequality of (4.3) and ∆ a ≥ q −
1, we get δ bi ≥ q −
1. Observe also that the useof Lemma 4.2 guarantees that all edges in H t b are still incident to vertices of P i .As done in a previous argument, if t b = t lk t b ′ , where l ≥ q − b ′ ∈ N s , then t b ∈ ( t ) + I ( X ),and so, we may assume that if t lk divides t b , then l ≤ q −
2. Since δ bi ≥ q −
1, there exist at leasttwo edges in H t b that have v i as an endpoint and such that the other endpoints are in V G \ ( P ∪ P i ).We now use Lemma 4.1 with v i and v instead of v and w . Then there exists a monomial t c ∈ S such that t b ≡ t c mod I ( X ), wt H tc ( v ) = wt H tb ( v ) >
0, and wt H tc ( v ) ≥ q − z ∈ V G \ ( P ∪ P i ) as one of the endpoints mentioned above ( t { v i ,z } dividing t b ), and suchthat t { v i ,z } is used in the “transfer of weight” from v i to v of Lemma 4.1. As a consequence, t { z,v } divides t c .If z = v , then t c ∈ ( t ), and we are done.Consider the case when z = v . Since wt H tc ( v ) >
0, there exists u ∈ V G such that t { v ,u } divides t c . If u = v , we are again done. If u = v , and since all edges in H t b are incident to vertices of P i ,we must have u ∈ P i .Now, H t c satisfies the assumptions of Subcase 1.1. We still need to consider the case when all edges { w , w } ∈ E ( H t a ) with w , w P are such that w = v i or w = v i . In this situation, δ ai = ∆ a ≥ q −
1, and we repeat the above argument, usingLemma 4.1 for t a instead of t b . Case 2.
Suppose that wt H ta ( v ) wt H ta ( v ) = 0.Assume, without loss of generality, that wt H ta ( v ) = 0 (if wt H ta ( v ) = 0, we argue in the sameway, exchanging v and v ). Then from (4.2) we get(4.4) P ri =2 wt H ta ( v i ) ≥ ( r − q −
2) + 2 . Denote by ∆ a the total weight of the edges in H t a between the vertices of V G \ P . By (4.1), α ( q −
2) + 1 ≤ d + 1 = P v ∈ P wt H ta ( v ) + ∆ a ≤ ( α − q −
2) + ∆ a , and thus we deduce that ∆ a ≥ q − Subcase 2.1.
Assume wt H ta ( v ) ≥ ( q −
1) + 1.Let { w , w } ∈ E ( H t a ) be an edge with w , w P and assume w , w = v . Then, Lemma 4.2gives a monomial t b such that t a − t b ∈ I ( X ), wt H tb ( v ) = wt H ta ( v ) = 0, wt H tb ( v ) = wt H ta ( v ) ≥ ( q −
1) + 1, and such that the total weight of the edges from v to the vertices of V G \ P , δ b =min (cid:8) wt H ta ( v ) , ∆ a (cid:9) , is ≥ q −
1. By Lemma 4.1 (with α = 1, and v and v instead of v and w ), we get a monomial t c such that t b − t c ∈ I ( X ), wt H tc ( v ) = wt H tb ( v ) − ( q − ≥ H tc ( v ) ≥ ( q − H t c satisfies the assumptions of Case 1.If all edges { w , w } ∈ E ( H t a ) with w , w P are such that w = v or w = v , then δ a = ∆ a ≥ q −
1, and repeating the argument (using Lemma 4.1 for t a instead of t b ), we fall again in Case 1. Subcase 2.2.
Suppose that 1 ≤ wt H ta ( v ) ≤ q −
1. Then (4.4) implies that there exists i ∈ { , . . . , r } such that wt H ta ( v i ) ≥ q −
1. We argue as in the previous subcase.Let { w , w } ∈ E ( H t a ) be an edge with w , w P and assume w , w = v i . Then, Lemma 4.2 givesa monomial t b such that t a − t b ∈ I ( X ), wt H tb ( v ) = wt H ta ( v ) = 0, wt H tb ( v ) = wt H ta ( v ) ≥ δ bi = min (cid:8) wt H ta ( v i ) , ∆ a (cid:9) ≥ q −
1. By Lemma 4.1 (with α = 1, and v i and v insteadof v and w ), we get a monomial t c such that t b − t c ∈ I ( X ), wt H tc ( v i ) = wt H tb ( v i ) − ( q − H tc ( v ) ≥ ( q − H tc ( v ) = wt H tb ( v ) ≥
1. Once again H t c satisfies the assumptions ofCase 1.If all edges { w , w } ∈ E ( H t a ) with w , w P are such that w = v i or w = v i , then δ ai = ∆ a ≥ q −
1, and repeating the argument (using Lemma 4.1 for t a instead of t b ), we fall again in Case 1. Subcase 2.3.
Suppose that wt H ta ( v ) = 0. Then, as in Subcase 2.2, there exists i ∈ { , . . . , r } suchthat wt H ta ( v i ) ≥ q −
1. Since wt H ta ( v ) = 0, we can repeat for ∆ a what we did for ∆ a : α ( q −
2) + 1 ≤ d + 1 = P v ∈ P wt H ta ( v ) + ∆ a ≤ ( α − q −
2) + ∆ a , and conclude that ∆ a ≥ q −
1. Using Lemmas 4.2 and 4.1, this time moving the weight of v i towards v , we can reduce to either Subcase 2.1 or Subcase 2.2. DEALS OVER COMPLETE MULTIPARTITE GRAPHS 15
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Rees algebras of edge ideals , Comm. Algebra. (1995), no. 9, 3513-3524.(Jorge Neves) CMUC, Department of Mathematics, University of Coimbra.Apartado 3008 - EC Santa Cruz, 3001-501 Coimbra, Portugal.
E-mail address : [email protected] (Maria Vaz Pinto) Departamento de Matem´atica, Instituto Superior T´ecnicoUniversidade T´ecnica de Lisboa, Avenida Rovisco Pais, 1, 1049-001 Lisboa, Portugal
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