Regularity of the vanishing ideal over a bipartite nested ear decomposition
aa r X i v : . [ m a t h . A C ] M a y REGULARITY OF THE VANISHING IDEALOVER A BIPARTITE NESTED EAR DECOMPOSITION
J. NEVES
Abstract.
We study the Castelnuovo–Mumford regularity of the vanishingideal over a bipartite graph endowed with a decomposition of its edge set.We prove that, under certain conditions, the regularity of the vanishing idealover a bipartite graph obtained from a graph by attaching a path of length ℓ increases by ⌊ ℓ ⌋ ( q − q is the order of the field of coefficients. We usethis result to show that the regularity of the vanishing ideal over a bipartitegraph, G , endowed with a weak nested ear decomposition is equal to | V G | + ǫ − ( q − , where ǫ is the number of even length ears and pendant edges of the decompo-sition. As a corollary, we show that for bipartite graph, the number of evenlength ears in a nested ear decomposition starting from a vertex is constant. Introduction
Given G , a simple graph, and K , a finite field, K [ E G ] denotes the polynomialring with coefficients in K , the variables of which are in one-to-one correspondencewith the edge set of the graph. The vanishing ideal over G is a binomial ideal of K [ E G ], denoted here by I q ( G ), given as the vanishing ideal of the projective toricsubset parameterized by E G . They were defined by Renteria, Simis and Villarrealin [16], with a view towards applications to the theory of linear codes and hencethe presence of a finite field. The aim of this work is to continue the study of theCastelnuovo–Mumford regularity of these ideals. Originally this invariant is relatedto the error-correcting performance of the linear codes involved, however, here, wewish to regard it strictly from the point of view of the link that these ideals providebetween commutative algebra and graph theory.This idea has been used for many classes of ideals and the existing results pointto interesting graph invariants. For instance, the Castelnuovo–Mumford regularityof the edge ideal of graph is bounded below by the induced matching number andabove by the co-chordal cover number (cf. [7], [8, Lemma 2.2] and [19]). There arealso partial results for the toric ideal of a graph (cf. [18, 1]) and for the binomialedge ideal of a graph (cf. [2, 9, 12]).The fact that, for the vanishing ideal over a graph, the quotient of K [ E G ] by I q ( G ) is a Cohen–Macaulay graded ring of dimension one explains why we knowrelatively more about this invariant in this case than in the cases of edge, toric or Mathematics Subject Classification.
Key words and phrases.
Castelnuovo–Mumford regularity, Binomial ideal, ear decomposition.This work was partially supported by Centro de Matem´atica da Universidade de CoimbraUID/MAT/00324/2013, funded by the Portuguese Government through FCT/MEC and co-fundedby the European Regional Development Fund through the Partnership Agreement PT2020. binomial edge ideals. The Castelnuovo–Mumford regularity of the vanishing idealover a graph has been computed for many classes of graphs, including trees, cycles(cf. [14]), complete graphs (cf. [6]), complete bipartite graphs (cf. [4]), completemultipartite graphs (cf. [13]) and, more recently, parallel compositions of paths(cf. [11]). Additionally we know that, in the bipartite case, the regularities of thevanishing ideals over the members of the block decomposition of a graph completelydetermine the regularity of the vanishing ideal over the graph (see Proposition 2.10,below).In this work we establish a formula for the Castelnuovo–Mumford regularityof the vanishing ideals over graphs in the class of bipartite graphs endowed withcertain decomposition of its edge set into paths. The simplest case of a such adecomposition, a so-called ear decomposition , is a partition of the edge set of G into subgraphs P , P , . . . , P r such that P is a vertex and, for all 1 ≤ i ≤ r ,the path has its end-vertices in P ∪ · · · ∪ P i − and none of its inner vertices inthis union. Ear decompositions play a central role in graph connectivity as, byWhitney’s theorem, a graph is 2-vertex-connected if and only if it is endowed withan open ear decomposition (one in which every P i with i > nested ear decomposition, aspecial case of ear decomposition in which, firstly, the paths P i are forced to haveend-vertices in a (same) P j , for some j < i , and, secondly, a nesting condition is tobe satisfied for two paths having their end-vertices in a same P j (see Definition 4.1).By Theorem 4.4, below, it follows that the Castelnuovo–Mumford of the quotientof K [ E G ] by I q ( G ), when G is endowed with a nested ear decomposition, is givenby:(1) | V G | + ǫ − ( q − ǫ is the number of paths of even length in the decomposition and q is theorder of the field K . As a corollary, we deduce that the number of even length pathsin any nested ear decomposition of a bipartite graph, that starts from a vertex, isconstant (cf. Corollary 4.5).This article is organized as follows. In the next section we set up the notationused throughout and define the vanishing ideal over a graph. We recall severalcharacterizations of this ideal which allow a direct definition without mentioningthe projective toric subset parameterized by the edges of the graph. We also recallthe Artinian reduction technique, which is the main tool in the computation ofthe regularity (cf. Proposition 2.5). After reviewing some known values of theregularity (cf. Table 1) we go through the existing results bounding the regularityin terms of combinatorial data on the graph, among these, the bound from theindependence number of the graph (cf. Proposition 2.6). Other results reviewedin this section include the upper bounds obtained from a spanning subgraph andfrom an edge cover and two other results, one relating the regularity with the blockdecomposition and another relating it with the leaves of the graph. Sections 3 and 4contain the main results of this work. In Section 3, we investigate the contributionto the regularity of the vanishing ideal over a graph obtained from another graphby attaching to it a path by its end-vertices. Theorem 3.4 states that, under someconditions, the regularity increases by ⌊ ℓ ⌋ ( q − ℓ is the length of thepath attached and q the cardinality of K . In Section 4, we use the previous result EGULARITY OF VANISHING IDEALS 3 to establish the regularity of a bipartite graph endowed with a weak nested eardecomposition. This notion is a slight generalization of the notion of nested eardecomposition and arises naturally in the context of the proof of Theorem 4.4.Its distinctive feature is that one allows the existence of pendant edges in thedecomposition. Theorem 4.4 expresses the regularity of a bipartite graph endowedwith a weak nested ear decomposition by the formula (1), where, now, ǫ is thenumber of even length paths and pendant edges. Corollary 4.5, stating that thenumber of even length paths in a nested ear decomposition of a graph is constant,is then a direct consequence of this formula. As an application of Theorem 4.4 wefinish by producing a family of graphs with regularities arbitrarily larger than thelower bound given by their independence numbers.2. Preliminaries
The graphs considered in this work are assumed to be simple graphs (finite,undirected, loopless and without multiple edges). Additionally, we will assumethroughout that no isolated vertices occur. To simplify the notation, we assumethat the vertex set, V G , is a subset of N .2.1. The vanishing ideal over a graph.
We will denote by K a finite field oforder q >
2. Given a graph G , we consider a polynomial ring with coefficients in K the variables of which are in bijection with the edges of G and denote it by K [ E G ].A variable in K [ E G ] corresponding to and edge { i, j } ∈ E G will be denoted by t ij , which is the abbreviated form of t { i,j } . Given an non-negative integer valuedfunction on the edge set, α ∈ N E G , the monomial t α ∈ K [ E G ] is, by definition, t α = Y { i,j }∈ EG t α { i,j } ij . We say that t α is supported on the edges of a subgraph H ⊂ G if α { i, j } 6 = 0 ⇐⇒ { i, j } ∈ E H . Consider P | E G |− , the projective space over K with coordinate ring K [ E G ] and let P | V G |− , be the projective space with coordinate ring K [ x i : i ∈ V G ]. The ringhomomorphism ϕ : K [ E G ] → K [ x i : i ∈ V G ] given by:(2) t ij x i x j defines a rational map ϕ ♯ : P | V G |− → P | E G |− , the restriction of which to the pro-jective torus, T | V G |− , the subset of projective space of points with every coordinatea nonzero scalar, is a regular map. Definition 2.1.
The projective toric subset parameterized by G is the subset of P | E G |− defined by: X = ϕ ♯ ( T | V G |− ) ⊂ P | E G |− . The vanishing ideal of X is denoted by I q ( G ) ⊂ K [ E G ].We note that I q ( G ) can be defined directly from G , without reference to X , asthe ideal generated by the homogeneous polynomials f ∈ K [ E G ] which vanish aftersubstitution of each variable t ij by a i a j , for all a i ∈ K ∗ , with i ∈ V G . For thisreason we refer to I q ( G ) simply as the vanishing ideal over G .The ideal I q ( G ) was defined in [16]. Being a vanishing ideal, it is automaticallya radical, graded ideal. We also know that I q ( G ) has a binomial generating set. J. NEVES
The fact that I q ( G ) contains the vanishing ideal of the torus over the finite field K ,which is given by(3) I q = (cid:0) t q − ij − t q − kl : { i,j } , { k,l } ∈ E G (cid:1) , implies that the height of I q ( G ) is | E G | − K [ E G ] /I q ( G )is a one-dimensional graded ring. Additionally, since any monomial in K [ E G ] is aregular element in this quotient (since no variable vanishes on the torus), we deducethat K [ E G ] /I q ( G ) is Cohen–Macaulay. We refer the reader to [16, Theorem 2.1]for complete proofs of these statements.The ideal I q ( G ) can be related to the toric ideal of G , i.e., the ideal P G ⊂ K [ E G ]given by P G = ker ϕ , where ϕ : K [ E G ] → K [ x i : i ∈ V G ] is the map defined by (2).It can be shown (see [16, Theorem 2.5]) that(4) I q ( G ) = ( P G + I q ) : ( t ∗ ) ∞ , where I q is the vanishing ideal of the torus, given in (3), and by t ∗ we denote theproduct of all variables of the polynomial ring K [ E G ], t ∗ = Q { i,j }∈ EG t ij . The relation with the toric ideal (4) reinforces the idea, already expressed above,that I q ( G ) can be defined without any reference to the projective toric subset X parameterized by E G . Yet another way to do this is by a characterization of theset homogeneous binomials of I q ( G ), achieved by the following proposition. Theproof of this result can be found in [13, Lemma 2.3]. Proposition 2.2.
Let t ν − t µ ∈ K [ E G ] be a homogeneous binomial. Then t ν − t µ belongs to I q ( G ) if and only if, for all i ∈ V G , (5) X k ∈ N G ( i ) ν { i,k } ≡ X k ∈ N G ( i ) µ { i,k } (mod q − , where N G ( · ) denotes the set of neighbors of a vertex. With this characterization of I q ( G ) by means of a generating set of homogeneousbinomials satisfying (5), the following relation between the ideal I q ( G ) and thevanishing ideal over a subgraph of G is easy to prove. Corollary 2.3.
Let H be a subgraph of G . Then, under the inclusion of polynomialrings K [ E H ] ⊂ K [ E G ] , we have I q ( H ) = I q ( G ) ∩ K [ E H ] . Despite the multiple characterizations of I q ( G ), a complete classification of thesubgraphs of G that support binomials of a minimal binomial generating set of I q ( G ) is still lacking, for general G ; in contrast with the case of the toric ideal P G in which the binomials in a minimal generating set are in one-to-one correspondencewith the closed even walks on the graph.2.2. Castelnuovo–Mumford regularity.
Recall that if S is a polynomial ringand M is any graded S -module, the Castelnuovo–Mumford regularity of M is, bydefinition, reg M = max i,j { j − i : β ij = 0 } , where β ij are the graded Betti numbers of M . The Castelnuovo–Mumford regularityof K [ E G ] /I q ( G ) is thus an integer we can associate to any simple graph withoutisolated vertices. EGULARITY OF VANISHING IDEALS 5
Definition 2.4.
Let G be a simple graph without isolated vertices and K a finitefield. We define the Castelnuovo–Mumford regularity of G over the field K to bethe regularity of the quotient K [ E G ] /I q ( G ) and we denote it by reg G .Since K [ E G ] /I q ( G ) is a Cohen–Macaulay one-dimensional graded ring, its regu-larity coincides with its index of regularity , i.e., the least integer from which thevalue of the Hilbert function equals the value of the Hilbert Polynomial (cf. [18,Proposition 4.2.3]). Additionally, given that any monomial t δ ∈ K [ E G ] is a re-gular element of K [ E G ] /I q ( G ) and the quotient of K [ E G ] by the extended ideal,( I q ( G ) , t δ ), is a zero-dimensional graded ring with index of regularity equal toreg G + deg t δ , we get:reg G = min (cid:8) i : dim K (cid:0) K [ E G ] / ( I q ( G ) , t δ ) (cid:1) i = 0 (cid:9) − deg t δ . The idea of taking the Artinian quotient K [ E G ] / ( I q ( G ) , t δ ) to compute reg G isthe main ingredient in the proof of the next proposition, which will be used severaltimes in this article. See [11, Propositions 2.2 and 2.3] for a proof. Proposition 2.5.
Let G be a graph, t δ ∈ K [ E G ] a monomial and d a positiveinteger. Then, reg G ≤ d − deg( t δ ) if and only if for every monomial t ν of degree d there exists t µ , of degree d , suchthat t δ | t µ and t ν − t µ ∈ I q ( G ) . Graph invariants and the regularity.
Table 1 contains the values of theCastelnuovo–Mumford regularity of several families of graphs. The simplest casesare those of a tree and an odd cycle. These are the simplest cases because, inGraph reg G Tree with s edges ( | V G | − q − | V G | − q − | V G |− ( q − K n , n ≥ ⌈ ( n − q − / ⌉ Complete bipartite graph K a,b (max { a, b } − q − Table 1. both, X , the projective toric subset parameterized by E G , coincides with the torus(cf. [16, Corollary 3.8]) and therefore the ideal I q ( G ) is equal to the vanishing idealof the torus, I q , given in (3). The fact that I q is a complete intersection enables thestraightforward computation of the regularity. In [14] the case of an even cycle was J. NEVES dealt with. The cases of the complete graph and of the complete bipartite graphwere studied in [6] and [4], respectively.By now, there are many ways to produce estimates for the Castelnuovo–Mumfordregularity of a particular graph using combinatorial invariants of the graph. We be-gin by mentioning the lower bound obtained from the vertex independence numberof the graph.
Proposition 2.6 ([11, Proposition 2.7]) . If V ⊂ V G is a set of r independentvertices, such that the edge set of G − V is not empty, then reg G ≥ r ( q − . Since G has no isolated vertices, it follows from this result that(6) reg G ≥ ( α ( G ) − q − , where α ( G ) is the vertex independence number of G . However, as can be easilyseen by the values of the regularity of Table 1, this bound is not sharp if G isnon-bipartite or, even for a bipartite graph, if it fails to be 2-connected. As anapplication of Theorem 4.4, we shall give an infinite family of 2-connected bipartitegraphs for which the bound (6) is not sharp. (See Example 4.6.)The operation of vertex identification also yields lower bounds for the Castel-nuovo–Mumford regularity of G . The next result was proved in [11, Proposition 2.5]. Proposition 2.7.
Let v and v be two nonadjacent vertices of G and let H be thesimple graph obtained after identifying v with v . Then reg G ≥ reg H . Note that the identification of two vertices can create multiple edges. By simplegraph, in the statement, we refer to the graph obtained after the removal of allmultiple edges created.Bounds for the Castelnuovo–Mumford regularity of a graph can also be obtainedfrom its subgraphs. The next result follows from [17, Lemma 2.13].
Proposition 2.8.
Let H be a spanning subgraph of G which is non-bipartite if G is non-bipartite. Then reg G ≤ reg H . Reversing the roles of G and H , this result can also be used to produce lower boundsof the regularity. For instance, if G is bipartite and spans a K a,b thenreg G ≥ (max { a, b } − q − G is non-bipartite with | V G | ≥
4, thenreg G ≥ reg K | V G | = (cid:6) ( | V G |− q − (cid:7) . Another way to obtain upper bounds for the regularity of a graph is by using adecomposition of G into two subgraphs with, at least, one edge in common. Proposition 2.9 ([11, Proposition 2.6]) . If H and H are two subgraphs of G with a common edge and G = H ∪ H then reg G ≤ reg H + reg H . A graph is said 2-vertex-connected (or simply 2-connected) if | V G | ≥ G − v is connected for every v ∈ V G . Any graph can be decomposed into a setof edge disjoint subgraphs consisting of either isolated vertices, single edges (calledbridges) or maximal 2-connected subgraphs. This decomposition is called the blockdecomposition of the graph. In [15] the relation between the regularity of a bipartitegraph and the regularities of the members of its block decomposition was described. EGULARITY OF VANISHING IDEALS 7
Proposition 2.10 ([15, Theorem 7.4]) . Let G be a simple bipartite graph withoutisolated vertices and let G = H ∪ · · · ∪ H m be the block decomposition of G , then (7) reg G = P mk =1 reg H i + ( m − q − . The previous result does not hold if we drop the bipartite assumption. Thegraph in Figure 1 is a counterexample. We used
Macaulay2 , [10], to computeits Castelnuovo–Mumford for some values of the order of the ground field. For q ∈ { , , , , , , , , } , the regularity is given by the formula ⌈ q − / ⌉ .On the other hand, its block decomposition has three blocks; two triangles, ofregularity 2( q − q − Figure 1.
The next proposition gives an additive formula for the regularity of G withrespect to its leaves which holds for both bipartite and non-bipartite graphs. Proposition 2.11 ([11, Proposition 2.4]) . If v , . . . , v r are vertices of degree oneand G ♭ is the graph defined by G ♭ = G − { v , . . . , v r } then reg G = reg G ♭ + r ( q − . Proposition 2.10 motivates the study of the regularity of a general 2-connectedbipartite graph. In view of Whitney’s structure theorem for 2-connected graphs (seeSection 4) one is naturally drawn to the problem of assessing the change producedin the regularity when we attach a path to a graph. We will explore this idea inthe next two sections. 3.
Ears and Regularity
The aim of this section is to provide a relation between the Castelnuovo–Mumfordregularities of a graph and of the graph obtained by attaching a path by its end-vertices. The main theorem of this section, Theorem 3.4, states that the additionof such a path increases the regularity by ⌊ ℓ (cid:5) ( q − ℓ is the length of thepath. In this result, we assume that G is bipartite and that the end-vertices of thepath are identified with two vertices of the graph which, in turn, are connected inthe graph by a path the inner vertices of which have degree two. Both assumptionsare necessary (see Examples 3.5 and 3.6). Proposition 3.2 addresses a special casein which we can afford to drop the bipartite assumption.By a path, P ⊂ G , we mean a subgraph of G endowed with an order of itsvertices, v , v , . . . , v ℓ , where ℓ >
0, such that v , . . . , v ℓ are ℓ distinct vertices and E P consists of the ℓ distinct edges { v i , v i +1 } , i = 0 , . . . , ℓ −
1. If v = v ℓ , P is alsocalled a cycle. However, note that since we are assuming that the edges { v i , v i +1 } are distinct, the case ℓ = 2 and v = v is not allowed. The inner vertices of P are v , . . . , v ℓ − and the end-vertices of P are v and v ℓ . The set of inner vertices of P J. NEVES will be denoted by P ◦ ⊂ V G . The number of edges in P is called the length of P and will be denoted by ℓ ( P ). Definition 3.1.
A path
P ⊂ G is called an ear of G if all inner vertices of P havedegree two in G . If the end-vertices of P are distinct, P is called an open ear ifthey coincide, P is called a pending cycle. Proposition 3.2.
Let
P ⊂ G be an ear of G , of length ℓ > . Assume either: (i) ℓ is odd and the end-vertices of P are distinct and adjacent in G or (ii) ℓ is even and the end-vertices of P coincide.Denote the graph G − P ◦ by G ♭ and assume that G ♭ has no isolated vertices. Then (8) reg G = reg( G ♭ ) + (cid:4) ℓ (cid:5) ( q − . Proof.
Note that since ℓ > ℓ is even ℓ ≥
4, we get ℓ ≥
3. Withoutloss of generality, we may assume that P is the path in G given by (1 , . . . , ℓ + 1),if ℓ is odd or (1 , . . . , ℓ,
1) if ℓ is even. (See Figure 2.) In both cases, it follows ······ ℓ +11 · · · ℓ ℓ − P (b) ······ · · · ℓ ℓ − P (a) Figure 2. that G contains an even cycle the vertex set of which coincides with V P . Since thegenerators of P G , the toric ideal of G , are given by the closed even walks on G ,using the relation between P G and I q ( G ), expressed in (4), it follows that(9) t t · · · t ℓ ( ℓ +1) − t t · · · t ( ℓ +1)1 ∈ I q ( G ) , if ℓ is odd, or t t · · · t ( ℓ − ℓ − t t · · · t ℓ ∈ I q ( G ) , if ℓ is even.Let us start by showing thatreg G ≥ reg G ♭ + (cid:4) ℓ (cid:5) ( q − t kl ∈ E G ♭ . By Proposition 2.5, applied to the graph G ♭ , we deduce that thereexists t α ∈ K [ E G ♭ ], of degree reg( G ♭ ), for which no monomial t β ∈ K [ E G ♭ ] divisibleby t kl is such that t α − t β ∈ I q ( G ♭ ).Let t ν ∈ K [ E G ] be the monomial of degree reg( G ♭ ) + (cid:4) ℓ (cid:5) ( q −
2) given by: t ν = t α ( t t · · · t ( ℓ − ℓ ) q − , if ℓ is odd, or t ν = t α ( t t · · · t ℓ ) q − , if ℓ is even.Suppose there exists t µ ∈ K [ E G ], with µ { k,l } >
0, such that(10) t ν − t µ ∈ I q ( G ) . EGULARITY OF VANISHING IDEALS 9
Modifying appropriately, with the use of t q − ij − t q − kl ∈ I q ( G ), we may assume that0 ≤ µ { i,i +1 } ≤ q −
2, for all i = 1 , . . . , ℓ −
1, and 0 ≤ µ { ℓ,ℓ +1 } ≤ q −
2, if ℓ isodd, or 0 ≤ µ { ,ℓ } ≤ q −
2, if ℓ is even. In other words, we may assume that thevariables along the path P appear in t µ raised to powers not greater than q − , . . . , ℓ we get,if ℓ is odd, µ { i − ,i } + µ { i,i +1 } ≡ q − , ∀ i ∈{ ,...,ℓ } or, if ℓ is even, µ { i − ,i } + µ { i,i +1 } ≡ q − , ∀ i ∈{ ,...,ℓ − } , and µ { ℓ − ,ℓ } + µ { ℓ, } ≡ q − . where all congruences are modulo q −
1. We deduce that there exist a, b ∈ { , . . . , q − } , with a + b ≡ q − ℓ is odd, ( µ { , } = µ { , } = · · · = µ { ℓ,ℓ +1 } = aµ { , } = µ { , } = · · · = µ { ℓ − ,ℓ } = b or, if ℓ is even, ( µ { , } = µ { , } = · · · = µ { ℓ − ,ℓ } = aµ { , } = µ { , } = · · · = µ { ℓ, } = b. Let t δ ∈ K [ E G ♭ ] be the monomial supported on G ♭ , given by t µ = ( t t · · · t ℓ ( ℓ +1) ) a ( t t · · · t ( ℓ − ℓ ) b t δ , if ℓ is odd, or t µ = ( t t · · · t ( ℓ − ℓ ) a ( t t · · · t ℓ ) b t δ , if ℓ is even.In view of (9), we deduce that there exists t β ∈ K [ E G ♭ ] such that β { k,l } ≥ µ { k,l } > t µ − ( t t · · · t ( ℓ − ℓ ) q − t β ∈ I q ( G ) , if ℓ is odd, or t µ − ( t t · · · t ℓ ) q − t β ∈ I q ( G ) , if ℓ is even.Note that if a + b > q − t q − ij − t q − kl ∈ I q ( G ), the powers of the variables t , t , . . . , t ( ℓ − ℓ can be reduced to q −
2. Combining (10) and (11) we obtain t α ( t t · · · t ( ℓ − ℓ ) q − − ( t t · · · t ( ℓ − ℓ ) q − t β ∈ I q ( G ) , if ℓ is odd, or t α ( t t · · · t ℓ ) q − − ( t t · · · t ℓ ) q − t β ∈ I q ( G ) , if ℓ is even.Since any product of variables is regular in K [ E G ] /I q ( G ), we get t α − t β ∈ I q ( G ) , where, recall β { k,l } >
0. But this binomial, being supported on G ♭ , also belongs to I q ( G ♭ ). This contradicts the assumptions on t α . Therefore, by Proposition 2.5,reg G ≥ deg( t ν ) = reg( G ♭ ) + (cid:4) ℓ (cid:5) ( q − . To prove the opposite inequality we will use Proposition 2.9. If ℓ is odd (seeFigure 2a), consider the decomposition of G given by P ∪ { , ℓ + 1 } and G ♭ . Then,reg G ≤ reg G ♭ + reg( P ∪ { , ℓ + 1 } ) = reg G ♭ + (cid:4) ℓ (cid:5) ( q − . If ℓ is even (see Figure 2b), we consider the decomposition of G into the subgraph H = G ♭ ∪ { , } and the cycle P . Using Propositions 2.9 and 2.11, we getreg G ≤ reg G ♭ + ( q −
2) + reg P = reg G ♭ + ℓ ( q − . (cid:3) Definition 3.3.
Let G be a bipartite graph and I ⊂ G an open ear of G . Abipartite ear modification of G along I is the simple graph obtained by either: (i)replacing I by another open ear P , with the same end-vertices and length of thesame parity as ℓ ( I ), or (ii), if ℓ ( I ) is even, by identifying the end-vertices of I in G − I ◦ . We say that G satisfies the bipartite ear modification hypothesis on I if,whenever G ′ is a bipartite ear modification of G along I , we have(12) reg G ′ = reg G + | V G ′ |−| V G | ( q − . Notice that, since G is assumed to be bipartite, if ℓ ( I ) is even, then its end-vertices are not adjacent and in the bipartite ear modification described in (ii),no loop is created. However, in both cases, to obtain a simple graph it may benecessary to remove the multiple edges created.It is easy to see that an even cycle satisfies the bipartite ear modification as-sumption on any of its open ears. Given that the regularity of a tree on n verticesis ( n − q − Theorem 3.4.
Let G be a bipartite graph and I and P be two open ears of G sharing the same end-vertices. Let G ♭ denote the graph G − P ◦ , if ℓ ( P ) > ,or G \ E P , if ℓ ( P ) = 1 . Assume that G ♭ satisfies the bipartite ear modificationhypothesis on I . Then, reg G = reg G ♭ + (cid:4) ℓ ( P )2 (cid:5) ( q − . Proof.
Note that, since G is bipartite, the lengths of I and P have the same parity.We may assume, without loss of generality, that P is the path (1 , . . . , ℓ + 1), where ℓ = ℓ ( P ) and I is the path ( ℓ + 1 , . . . , ℓ , ℓ = ℓ ( P ) + ℓ ( I ), as illustratedin Figure 3. ℓ +12 3 ℓ PI ℓ Figure 3.
If the vertices 1 and ℓ + 1 are neighbors in G ♭ then, by Proposition 3.2, theresult holds. Assume ℓ ( P ) = 1 and ℓ ( I ) >
1. Then the graph G − I ◦ is isomorphicto a bipartite ear modification of G ♭ and, accordingly,reg( G − I ◦ ) = reg G ♭ − ⌊ ℓ ( I )2 (cid:5) ( q − . EGULARITY OF VANISHING IDEALS 11
On the other hand, using again Proposition 3.2 we getreg G = reg( G − I ◦ ) + ⌊ ℓ ( I )2 (cid:5) ( q −
2) = reg G ♭ . Thus, from now on, we may assume that the vertices 1 and ℓ + 1 are not neighborsin G ♭ and ℓ ( I ) , ℓ ( P ) >
1. We will split the proof into two cases according to theparity of ℓ ( I ).We start by assuming that ℓ ( I ) is odd. Consider the graph G ♯ obtained byadding the edge E = { , ℓ + 1 } to G . Then, as G is a spanning subgraph of G ♯ ,we have reg G ≥ reg G ♯ . Denote the graph ( G ♭ − I ◦ ) ∪ E by ( G ♭ ) ′ . (See Figure 4.) ℓ ℓ +1 ℓ Figure 4.
The graph ( G ♭ ) ′ .Since ( G ♭ ) ′ is a bipartite ear modification of G ♭ along I ,reg( G ♭ ) ′ = reg G ♭ − (cid:4) ℓ ( I )2 (cid:5) ( q − . On the other hand, using Proposition 3.2,reg G ♯ = reg( G ♭ ) ′ + (cid:0)(cid:4) ℓ ( P )2 (cid:5) + ⌊ ℓ ( I )2 (cid:5)(cid:1) ( q − G ≥ reg G ♯ = reg G ♭ + (cid:4) ℓ ( P )2 (cid:5) ( q − . Let us now prove the opposite inequality. We will use induction on ℓ ( P )+ ℓ ( I )2 .Consider the following monomials in K [ E G ]:(13) t δ = t t · · · t ( ℓ − ℓ , t ǫ = t t · · · t ℓ ( ℓ +1) , t δ = t ( ℓ +2)( ℓ +3) · · · t ( ℓ − ℓ , t ǫ = t ( ℓ +1)( ℓ +2) · · · t ℓ . The monomial t δ is the monomial given by the multiplication of the variablesassociated to every other edge of P starting from the second. The monomial t ǫ isthe monomial given by the multiplication of the other edges of P . The monomials t δ and t ǫ are described similarly with respect to I . (See Figure 5.)Notice that deg( t δ ) = (cid:4) ℓ ( P )2 (cid:5) , deg( t ǫ ) = (cid:4) ℓ ( P )2 (cid:5) + 1 , deg( t δ ) = (cid:4) ℓ ( I )2 (cid:5) , deg( t ǫ ) = (cid:4) ℓ ( I )2 (cid:5) + 1 . t δ ℓ ℓ +1 ℓ t ǫ ℓ ℓ +1 ℓ t δ ℓ ℓ +1 ℓ t ǫ ℓ ℓ +1 ℓ Figure 5.
Edges in the support of t δ , t ǫ , t δ and t ǫ .Also, since t ǫ t δ − t δ t ǫ is a generator of the toric ideal of the even cycle P ∪ I ,(14) t ǫ t δ − t δ t ǫ ∈ I q ( P ∪ I ) ⊂ I q ( G ) . By Proposition 2.5, to show that(15) reg G ≤ reg G ♭ + (cid:4) ℓ ( P )2 (cid:5) ( q − t ν ∈ K [ E G ] of degreereg G ♭ + (cid:4) ℓ ( P )2 (cid:5) ( q −
2) + (cid:4) ℓ ( P )2 (cid:5) + (cid:4) ℓ ( I )2 (cid:5) there exists t µ ∈ K [ E G ], of the same degree as t ν , divisible by t δ t δ , such that t ν − t µ belongs to I q ( G ). Set t ν = t α t β t γ for some α, β, γ ∈ N E G satisfying t α ∈ K [ E P ], t β ∈ K [ E I ], t γ ∈ K [ E G ♭ −I ◦ ]. Then,(16) deg( t α ) + deg( t β ) + deg( t γ ) = reg G ♭ + (cid:4) ℓ ( P )2 (cid:5) ( q −
1) + (cid:4) ℓ ( I )2 (cid:5) . Suppose that(17) deg( t α ) ≥ (cid:4) ℓ ( P )2 (cid:5) ( q −
1) and deg( t β ) ≥ (cid:4) ℓ ( I )2 (cid:5) ( q − . Then, t α t β , which is supported on the cycle P ∪ I , is such thatdeg( t α t β ) ≥ reg( P ∪ I ) + ⌊ ℓ ( P )2 (cid:5) + ⌊ ℓ ( I )2 (cid:5) and then, by Proposition 2.5, applied to P ∪ I , there exists t µ ∈ K [ E P∪I ], divisibleby t δ t δ such that t α t β ∈ I q ( P ∪ I ) ⊂ I q ( G ). We deduce that t α t β t γ − t µ t γ ∈ I q ( G ) , as desired. Assume now that (17) does not hold. Now, directly from (16),deg( t α ) < (cid:4) ℓ ( P )2 (cid:5) ( q − ⇐⇒ deg( t β t γ ) ≥ reg G ♭ + (cid:4) ℓ ( I )2 (cid:5) + 1 . On the other hand, since G − I ◦ is a bipartite ear modification of G ♭ along I , andtherefore by our assumptions,reg( G − I ◦ ) = reg G ♭ + (cid:4) ℓ ( P )2 (cid:5) ( q − − (cid:4) ℓ ( I )2 (cid:5) ( q − , we get from (16):deg( t β ) < (cid:4) ℓ ( I )2 (cid:5) ( q − ⇐⇒ deg( t α t γ ) ≥ reg( G − I ◦ ) + (cid:4) ℓ ( P )2 (cid:5) + 1 . EGULARITY OF VANISHING IDEALS 13
Hence, by symmetry, we may assume that(18) deg( t α ) < (cid:4) ℓ ( P )2 (cid:5) ( q − ⇐⇒ deg( t β t γ ) ≥ reg G ♭ + (cid:4) ℓ ( I )2 (cid:5) + 1 . Then, by Proposition 2.5, there exists t µ ∈ K [ E G ♭ ], of degree equal to deg( t β t γ ),divisible by t δ , such that t β t γ − t µ ∈ I q ( G ♭ ) ⊂ I q ( G ) , which implies that t α t β t γ − t α t µ ∈ I q ( G ) . If t δ divides t α we have finished. Assume t δ does not divide t α . If t ǫ divides t α ,then, since t α t µ = ( t α t − ǫ )( t ǫ t δ )( t − δ t µ )and t ǫ t δ − t δ t ǫ ∈ I q ( G ) we get:(19) t α t β t γ − ( t α t − ǫ t δ )( t ǫ t − δ t µ ) ∈ I q ( G ) , where, deg( t ǫ t − δ t µ ) = deg( t β t γ ) + 1 ≥ reg G ♭ + (cid:4) ℓ ( I )2 (cid:5) + 2 . Hence, by Proposition 2.5, there exists t ρ ∈ K [ E G ♭ ] divisible by t δ such that(20) t ǫ t − δ t µ − t ρ ∈ I q ( G ♭ ) ⊂ I q ( G ) . From (19) and (20), we deduce that t α t β t γ − ( t α t − ǫ t δ ) t ρ ∈ I q ( G )where ( t α t − ǫ t δ ) t ρ is divisible by t δ t δ , as required.We may assume from now on that t α is divisible by neither t ǫ nor t δ . Sinceshowing that there exists a monomial t µ ∈ K [ E G ] of degree equal to the degree of t ν = t α t β t γ , divisible by t δ t δ such that t ν − t µ ∈ I q ( G ) is, by [11, Lemma 2.1],equivalent to showing that the same holds for the monomial obtained by permutingthe variables of the support of t δ and permuting the variables of the support of t ǫ , we may assume that neither t nor t divides t ν .Consider the graph H obtained from G by removing the edges { , } and { , } ,and identifying the vertices 1 and 3, as illustrated in Figure 6. Denote the earobtained from P after this operation by Q . By induction:reg H = reg( H − Q ◦ ) + (cid:4) ℓ ( Q )2 (cid:5) ( q −
2) = reg G ♭ + (cid:4) ℓ ( P )2 (cid:5) ( q − − ( q − . Let d = ν { , } and let t ν ∈ K [ E H ] be given by: t ν = t ν t − d t d . Then, since deg( t ν ) = deg( t ν ) = reg H + (cid:4) ℓ ( P )2 (cid:5) + (cid:4) ℓ ( I )2 (cid:5) + ( q − t δ t − t δ t q − ℓ ∈ K [ E H ] is such thatdeg( t δ t − t δ t q − ℓ ) = (cid:4) ℓ ( P )2 (cid:5) + (cid:4) ℓ ( I )2 (cid:5) + ( q − , by Proposition 2.5 applied to the graph H , there exists t µ ∈ K [ E H ] such that t δ t − t δ t q − ℓ divides t µ and t ν − t µ ∈ I q ( H ). Let c = µ { , } and let t µ ∈ K [ E G ]be given by: t µ = t µ t − c t c . ℓ +12 3 ℓ ℓ G ℓ +14 ℓ ℓ H Figure 6.
By Proposition 2.2, the binomial t ν − t µ satisfies a set of congruences modulo q − H . In particular, at 1 ∈ V H , we have:(21) d + X k ∈ N G (1) ν { ,k } = X k ∈ N H (1) ν { ,k } ≡ X k ∈ N H (1) µ { ,k } = c + X k ∈ N G (1) µ { ,k } . Let a ∈ { , . . . , q − } be such that a ≡ d − c and let b = ( q − − a . Then, as t δ t − t δ t q − ℓ divides t µ and hence it divides t µ , the binomial(22) t ν − t µ t − ( q − ℓ t b t a is a homogeneous binomial of the ring K [ E G ]. Moreover, since a ≥
1, we deducethat t δ t δ divides the monomial on the right side of (22).Let us prove that the binomial (22) belongs to I q ( G ). It suffices to check that thecorresponding congruences at vertices 1, 2 and 3 of G are satisfied, since at anyother vertex the corresponding congruence is identical to the one in H . At thevertices 2 and 3, we get, respectively,0 ≡ a + b and d ≡ c + a ⇐⇒ d − c ≡ a which hold, by the definitions of b and a . At the vertex 1, we get: X k ∈ N G (1) ν { ,k } ≡ b − ( q −
1) + X k ∈ N G (1) µ { ,k } = c − d + X k ∈ N G (1) µ { ,k } , which holds by (21). Hence (22) belongs to I q ( G ). This concludes the proof, byinduction, of the inequality (15) in the case of ℓ ( I ) odd.Let us now consider the case of ℓ ( I ) and ℓ ( P ) even. We start by proving that(23) reg G ≥ reg G ♭ + ℓ ( P )2 ( q − . Consider the simple graph H obtained from G by identifying the vertices 1 and ℓ + 1 and denote by H ′ be the subgraph of H obtained from G ♭ − I ◦ under thesame identification. (See Figure 7.)Since H ′ is a bipartite ear modification of G ♭ ,reg H ′ = reg G ♭ − ℓ ( I )2 ( q − . EGULARITY OF VANISHING IDEALS 151= ℓ +12 3 ℓ ℓ Figure 7.
On the other hand, using Propositions 2.7 and 3.2, or, in the case of ℓ ( P ) = 2 or ℓ ( I ) = 2, Proposition 2.11,reg G ≥ reg H = reg H ′ + ℓ ( P )2 ( q −
2) + ℓ ( I )2 ( q −
2) = reg G ♭ + ℓ ( P )2 ( q − , which proves (23). To prove that(24) reg G ≤ reg G ♭ + ℓ ( P )2 ( q − , by Proposition 2.5, it suffices to show that for every t ν ∈ K [ E G ] with(25) deg( t ν ) = reg G ♭ + ℓ ( P )2 ( q −
2) + 1 , there exists t µ , divisible by t , such that t ν − t µ ∈ I q ( G ). Consider the followinggraphs: ( G ♭ ) ∗ = G ♭ ∪ { , } , C = P ∪ I and G − I ◦ . (See Figure 8.) By Proposition 2.11, reg( G ♭ ) ∗ = reg G ♭ + ( q − C is an evencycle, reg C = ℓ ( P )2 ( q −
2) + ℓ ( I )2 ( q − − ( q − . Finally, since G ♭ satisfies the bipartite ear modification hypothesis,reg( G − I ◦ ) = reg G ♭ − ℓ ( I )2 ( q −
2) + ℓ ( P )2 ( q − . Let us write t ν = t α t β t γ for some α, β, γ ∈ N E G satisfying t α ∈ K [ E P ] , t β ∈ K [ E I ] and t γ ∈ K [ E G ♭ −I ◦ ] . Suppose that(26) deg( t α ) + deg( t β ) ≤ reg C deg( t β ) + deg( t γ ) ≤ reg( G ♭ ) ∗ deg( t α ) + deg( t γ ) ≤ reg( G − I ◦ )Then, deg( t ν ) ≤ (cid:2) reg C + reg( G ♭ ) ∗ + reg( G − I ◦ ) (cid:3) = reg G ♭ + ℓ ( P )2 ( q − , which is in contradiction with (25). Hence the opposite inequality of one of theinequalities in (26) must hold. For instance if it is the opposite of the first inequality,then, by Proposition 2.5, there exists t µ ∈ K [ E C ], divisible by t , such that t α t β − t µ ∈ I q ( C ) ⊂ I q ( G ) ( G ♭ ) ∗ C G − I ◦ Figure 8. which implies that t ν − t µ t γ ∈ I q ( G ) , as desired. We argue similarly for the other two cases. This proves (24) andconcludes the proof of the theorem. (cid:3) The next two examples show that the assumptions of Theorem 3.4 are strictlynecessary.
Example 3.5.
Let G ♭ be the graph of Figure 9. This graph decomposes into a cycleof length six and two cycles of length four. By Proposition 2.9, reg G ♭ ≤ q − V = { , , , } is an independent set for which G − V hasa nonempty edge set. Hence, by Proposition 2.6, reg G ♭ ≥ q − G ♭ = 4( q − G be the graph obtained by adding the edge { , } to G ♭ . Figure 9.
Unlike G ♭ , the graph G has now spanning cycle (of length 8). By Proposition 2.8,we get reg G ≤ q − G = 3( q − G ♭ = reg G , does not hold. This is because the hypothesis that, EGULARITY OF VANISHING IDEALS 17 besides the edge { , } , there should be another ear in G with end-vertices 2 and8, is not satisfied. Example 3.6.
Consider the graph, G , illustrated in Figure 10. G is a non-bipartiteparallel composition of paths; two of length two and a path of length three. Ac- Figure 10. cording to [11, Theorem 1.2], reg G = 4( q − G ♭ ⊂ G the subgraphgiven by the parallel composition of one of the paths of length two and the path oflength three. Then, G ♭ is a cycle of length 5 and, accordingly, reg G ♭ = 4( q − G ♭ to be the parallel composition of the two paths of length two. Thenreg G ♭ = q −
2. In both cases, the conclusion of Theorem 3.4 does not hold.4.
Nested Ear Decompositions
The goal of this section is to give a formula for the Castelnuovo–Mumford regu-larity of a graph endowed with a special decomposition into paths. An ear decom-position of a graph consists of a collection of r > P , P , . . . , P r , the edge sets of which form a partition of E G , such that P is a vertex and, forall 1 ≤ i ≤ r , P i is a path with end-vertices in P ∪ · · · ∪ P i − while none of itsinner vertices belong to P ∪ · · · ∪ P i − . The paths P , . . . , P r are called ears ofthe decomposition of G . We note that P i is not necessarily an ear of G , accordingto Definition 3.1, as its inner vertices may become end-vertices of the followingears. An ear decomposition is called open if all of paths P , . . . , P r have distinctend-vertices. It is well known that a graph is 2-vertex-connected if and only if ithas an open ear decomposition (Whitney’s Theorem). More generally, a graph is2-edge-connected if and only if has an ear decomposition. Definition 4.1.
Let P , P , . . . , P r be an ear decomposition of a graph, G . If apath P i has both its end-vertices in P j we say that P i is nested in P j and define thecorresponding nest interval to be the subpath of P j determined by the end-verticesof P i , if they are distinct, or, if they coincide, to be that single end-vertex. An eardecomposition of G is nested if, for all 1 ≤ i ≤ r , the path P i is nested in a previoussubgraph of the decomposition, P j , with j < i , and, in addition, if two paths P i and P l are nested in P j , with j < i, l , then either the corresponding nest intervalsin P j have disjoint edge sets or one edge set is contained in the other.Nested ear decompositions were introduced by Eppstein in [3]. In the originaldefinition P is allowed to be a path and thus the graphs considered in [3] are notnecessarily 2-edge-connected.The main result of this section is Theorem 4.4, which gives a formula for theCastelnuovo–Mumford regularity of a bipartite graph endowed with a nested ear decomposition. In the proof of this result we will need to show that a graph endowedwith a nested ear decomposition satisfies the bipartite ear modification hypothesisalong a certain ear. However, an instance of a bipartite ear modification, namelythe one involving removing the ear and identifying its end-vertices can modify theear decomposition structure by introducing pendant edges and, thus, producing agraph G ′ which may well not be 2-edge connected. We remedy this by working ona wider class of graphs, that of graphs endowed with a weaker form of nested eardecompositions. Definition 4.2. A weak nested ear decomposition of a graph is a collection ofsubgraphs P , . . . , P r , with r >
0, the edge sets of which form a partition of E G ,such that P is a vertex and, for every 1 ≤ i ≤ r , P i is a path such that either(i) both end-vertices of P i belong to some P j , with j < i and none of its innervertices belongs P ∪ · · · ∪ P i − , or(ii) if ℓ ( P i ) = 1, only one of the end-vertices of P i belongs to P ∪ · · · ∪ P i − .If P i has both its end-vertices in P j , the nest interval of P i in P j is defined asbefore. If ℓ ( P i ) = 1 and only one end-vertex belongs to a previous P j then the nestinterval is defined to be this vertex. The nesting condition of Definition 4.1 is thesame.If both end-vertices of P i belong to a previous P j , then P i will be referred to asan ear of the decomposition, otherwise, if ℓ ( P i ) = 1 and P i has only one end-vertexin a previous P j , then P i will referred to as a pendant edge of the decomposition.An ear with coinciding end-vertices will also be referred to as a pending cycle ofthe decomposition. Notice that when ℓ ( P i ) = 1, P i can either be an ear (so-called trivial ear) or a pendant edge of the decomposition. Figure 11 shows agraph that can be endowed with a weak nested ear decomposition. For instance,
12 3 45 67 98 10 12 11
Figure 11. P = 1, P = (1 , P = (1 , P = (1 , , , P = (3 , , , P = (7 , P = (8 , P = (9 , , , , P = 1, P = (1 , P = (1 , , , , EGULARITY OF VANISHING IDEALS 19
Lemma 4.3.
Let P , P , . . . , P r be a weak nested ear decomposition of a graph G .Then, there exists i > such that either P i is a pendant edge of G , or a pendantcycle of G , or an ear of G with distinct end-vertices such that for any P k containingboth end-vertices of P i , the subpath of P k induced by them is an ear of G .Proof. We argue by induction on r ≥
1. If r = 1 then it suffices to take i = 1. If r >
1, consider the graph G ♭ = P ∪ · · · ∪ P r − . By induction, there exist i > P i satisfying the conditions in the statement. If P r is pendant edge we may P r P i (a) P i P r (b) P i I l P l P r (c) Figure 12. take i = r for G . The same applies if P r is a pending cycle. Assume then that P r has distinct end-vertices. If P i is a pendant edge of G ♭ and it ceases to be so in G ,then the end-vertices of P r must coincide with the end-vertices of P i and the only P k that contain these vertices are then P r and P i which are both ears of G . (SeeFigure 12a.) In this case, P r satisfies the conditions for G . If P i is a pending cycleof G ♭ which ceases to be an ear of G then one of the end-vertices of P r must be aninner vertex of P i . Then, arguing as before, we see that P r satisfies the conditions.(See Figure 12b.) Finally, assume that P i is an ear of G ♭ with distinct end-vertices.Let I , . . . , I r − be the subpaths induced by the end-vertices of P i in the paths P , . . . , P r − . For ease of notation consider I j equal to the empty set if P j doesnot contain both end-vertices of P i . If the end-vertices of P r do not coincide withany inner vertex of the paths I i , . . . , I r − then P i satisfies the conditions of thestatement for G . Assume that an end-vertex of P r is an inner vertex of I l . Then, as I l is an ear of G ♭ , P r has to be nested in P l . Since P i is also nested in P l the nestintervals must be nested. This means that the subpath induced by the end-verticesof P r in P l must be contained in I l . (See Figure 12c.) Then, P r satisfies theconditions of the statement for G as the only paths that contain the end-verticesof P r are then P r and P l . (cid:3) Theorem 4.4.
Let P , P , . . . , P r be a weak nested ear decomposition of a bipartitegraph, G . Let ǫ denote the number of even ears and pendant edges of the decompo-sition. Then, (27) reg G = | V G | + ǫ − ( q − . Proof.
We will argue by induction on r ≥
1. If r = 1 then G is either an even cycleor a single edge. In both cases ǫ = 1 and (27) gives reg G = | V G |− ( q − G = 0, in the case of the edge. Both are correct.Assume that (27) holds for any bipartite graph endowed with a weak nested eardecomposition with r paths and consider G a graph endowed with a weak nestedear decomposition P , . . . , P r +1 with r + 1 paths. Throughout the remainder of theproof, denote by ǫ the number of even ears and pendant edges of this decomposition.By Lemma 4.3, there exist i >
0, such that P i is either a pendant edge, or a pendantcycle, or an ear of G with distinct end-vertices such that for any P k containing bothend-vertices of P i , the subpath of P k induced by them is an ear of G . It is clearthat in any of the cases(28) G ♭ = P ∪ · · · ∪ P i − ∪ P i +1 ∪ · · · ∪ P r +1 is a bipartite graph endowed with a weak nested ear decomposition. If P i is apendant edge of G , then by Proposition 2.11 and induction,reg G = reg G ♭ + ( q −
2) = | V G |− ǫ − − ( q −
2) + ( q −
2) = | V G | + ǫ − ( q − . If P i is pendant cycle of G , then ℓ ( P i ) is even and, by induction and Proposition 3.2,reg G = | V G |− ℓ ( P i )+1+( ǫ − − ( q −
2) + ℓ ( P i )2 ( q −
2) = | V G | + ǫ − ( q − . Assume that P i has distinct end-vertices and that for any P k containing both end-vertices of P i , the subpath of P k induced by them is an ear of G . Denote theend-vertices of P i by v and w . If v and w are adjacent in G then ℓ ( P i ) must beodd. Accordingly, by induction and Proposition 3.2,reg G = | V G |− ℓ ( P i )+1+ ǫ − ( q −
2) + ℓ ( P i ) − ( q −
2) = | V G | + ǫ − ( q − . Assume now that v and w are not adjacent in G and let j > P i is nested in P j . By the minimality of j one of v or w , say v ,belongs to P ◦ j . Denote the nest interval of P i in P j by I . (See top of Figure 13.)To be able to use Theorem 3.4, it will now suffice to show that G ♭ satisfies thebipartite ear modification hypothesis on I . If G ′ is a bipartite ear modificationof G ♭ along I , which does not involve the identification of the end-vertices of I ,then, as v and w are not adjacent, no multiple edges arise and the weak nested eardecomposition of G ♭ induces a weak nested ear decomposition of G ′ in which theonly change is in the length of P j , which, nevertheless, remains of the same parity.By induction, we can use (27) on both G ♭ and G ′ and the only change will be onthe cardinality of the sets of vertices of these graphs. It follows thatreg G ′ = reg G ♭ + | V G♭ |−| V G ′ | ( q − , EGULARITY OF VANISHING IDEALS 21 P k P iv w P j I P ′ kv = w P ′ j Figure 13. which is condition (12) of the bipartite ear modification hypothesis.Suppose now that ℓ ( I ) is even and that G ′ is obtained by identifying the end-vertices of I in G ♭ − I ◦ and removing the multiple edges created. For k = i , let P ′ k ⊂ G ′ denote the graph obtained by identifying v with w in P k −I ◦ and removingthe multiple edges created. (See Figure 13.) We note that since I is an ear of G , for k = j , the graph P ′ k is obtained by simply identifying v and w in P k and removingall multiple edges created. P k is isomorphic to P ′ k if one of v or w does not belongto P k . If both v and w belong to P k then, by the minimality of j , we must have j < k . But then none of v or w can be an inner vertex of P k . Accordingly, theymust coincide with the end-vertices of P k . (See top of Figure 13.) Then, I ∪ P k is a cycle of G and, since it must be of even length, we deduce that ℓ ( P k ) is alsoeven. In this situation P ′ k is either a pending (even) cycle of G ′ , if ℓ ( P k ) >
2, ora pending edge, if ℓ ( P k ) = 2. (See the bottom of Figure 13.) As for P ′ j , it maybe a single edge if ℓ ( P j ) − ℓ ( I ) = 1 or if ℓ ( P j ) − ℓ ( I ) = 2 and P j has coincidingend-vertices.It is clear that(29) G ′ = P ∪ P ′ ∪ · · · ∪ P ′ i − ∪ P ′ i +1 ∪ · · · ∪ P ′ r +1 , as any edge in G ′ comes from an edge in some path P k . If (29) does not induce apartition of the edge set of G ′ , there exist a vertex u and two edges { u, v } , { u, w } belonging to different paths, which become the same edges after the identification of v with w . Consider the least k for which the path P k contains both vertices u, v andthe least l for which P l contains u, w . We claim that P j = P k , and, consequently, u must belong to P j .Assume, to the contrary, that j = k . Since v ∈ P ◦ j we have j < k and then v is an end-vertex of P k . If u in an inner vertex of P k then we must have k ≤ l .If u is an end-vertex of P k then, by the minimality of k , P k has to be a pending edge and we get the same conclusion, k ≤ l . Assume that k = l . Then j < k = l implies that P k = P l is a path with end-vertices v and w , containing u as an interiorpoint. However if { u, v } and { u, w } belong to different paths then the degree of u isnot two, which contradicts the assumption on v and w , stating that these verticesinduce on any path of the weak decomposition a subpath the inner vertices of whichhave degree two. Hence we must have k < l . Then j < k < l implies that P l is theedge { u, w } . Since both its end-vertices belong to earlier paths, this contradicts theminimality of l .We have proved that P j = P k and, in particular, that u ∈ P j . Resetting notation,Let now P k and P l be any two (distinct) paths containing the edges { u, v } and { u, w } , respectively. We claim that either P k is a non-pending odd path of theweak ear decomposition of G ♭ and P ′ k is an edge or P l is a non-pending odd pathof the weak ear decomposition of G ♭ and P ′ l is an edge.To see this, we start by noting that, since v ∈ P ◦ j , we have j ≤ k . If j < k , then as u and v belong to P j they must be end-vertices of P k . In this case, P k = { u, v } ,which is a non-pending odd path and P ′ k an edge. If j = k then there are twosub-cases. Either j = k < l and then P l = { u, w } , which is a non-pending odd earwith P ′ l an edge (see Figure 14a), or we have l < j = k and then u and w mustbe the end-vertices of P j = P k . This implies that ℓ ( P j ) = ℓ ( I ) + 1, which is a oddinteger, and that P ′ j is an edge. (See Figure 14b.) u v w P j = P k P l (a) v u w P k = P j P l (b) Figure 14.
We conclude that, for (29) to induce a partition of the edge set of G ′ it sufficesto remove appropriately from (29) the paths that consist of a single repeated edgecoming from paths as described above. We note that, in this situation, the numberof even ears and pending edges, after removing all repeated edges in (29), coincideswith the number of even ears and pending edges of the weak nested decompositionof G ♭ given by (28).Let us now prove that, after the exclusion from (29) of the repeated edges, we obtaina weak nested ear decomposition of G ′ . Since I is an ear of G , it is clear that theend-vertices of P ′ k belong to P ′ l , for some l < k . An inner vertex of P ′ k alwayscomes from an inner vertex of P k . If, following the bipartite ear modification, suchvertex is identified with a vertex of a previous P ′ l , then the two vertices in questionmust v and w . We deduce that P k = P j . If w is also an inner vertex of P j then v = w in P ′ j cannot belong to any earlier P ′ l . If w is an end-vertex of P j then v = w becomes an end-vertex of P ′ j . As for the nesting condition, let P ′ k and P ′ l be two paths nested in P ′ s , with s < k, l . We may assume that none of P ′ k or P ′ l isa pending edge or cycle for otherwise there will be nothing to show. Let v = v be the end-vertices of P ′ k and w = w those of P ′ l . If the nest intervals of P ′ k and EGULARITY OF VANISHING IDEALS 23 P ′ l in P ′ s have inner vertices in common and are not nested, then, in the orderingof vertices of P ′ s , we must have, without loss of generality, v < w < v < w .This is impossible before the identification of v with w , since we are starting froma weak nested ear decomposition of G ♭ . Hence both v and w belong to P s and oneof w or v must be the vertex obtained by identifying v with w . Assume, withoutloss of generality, that this vertex is w . Then v , v come from P s unchanged andone of them is an inner vertex of the subpath induced by v and w in P s , but thisis impossible since, by assumption, v and w induce on P s a subpath whose innervertices have degree two in G .Having established that, after removing all redundant edges, (29) induces a weaknested ear decomposition of G ′ we can now use induction to compute its regularity.Since the fact that ℓ ( I ) is even implies that ℓ ( P i ) is even, we deduce that thenumber of even ears and pending edges of the weak decomposition of G ♭ given in(28) is ǫ −
1. Accordingly,reg G ′ = | V G ′ | +( ǫ − − ( q −
2) = reg G ♭ + | V G ′ |−| V G♭ | ( q − , and therefore, G ♭ satisfies the bipartite ear modification assumption along I . Thisfinishes the proof of the theorem. (cid:3) The number of even length paths in an ear decomposition of a graph is not nec-essarily constant, even if we restrict to bipartite graphs. Take, for example, thegraph obtained from the graph in Figure 9 by adding the edge { , } . As a firstear decomposition, consider the one obtained by starting from vertex 1 and addingconsecutively the paths (1 , , , , , , , , , ,
7) and (2 , , , , , , , , , , ,
8) and (2 , Corollary 4.5.
Let G be a bipartite graph endowed with a weak nested ear decom-position. Then the number of even ear and pendant edges of any weak nested eardecomposition of G remains constant. We will finish by giving another application of Theorem 4.4. As mentioned inSection 2, it follows from Proposition 2.6 thatreg G ≥ ( α ( G ) − q − , where α ( G ) denotes the independence number of G . This bound is not sharp if G is not bipartite or 2-connected, but equality does hold if G is an even cycle, ora complete bipartite graph, or a bipartite parallel composition of paths (see [11]),among many other examples. This could suggest that for a bipartite 2-connectedgraph the Castelnuovo–Mumford regularity of a graph is closely related with α ( G ).In the following example we want to show that this is not the case. Example 4.6.
Fix an even positive integer k . Consider the graph G , in Figure 15,below, obtained from a cycle of length 3 k , by attaching k ears of length two at thepairs of vertices 3 i − i , for each i = 1 , . . . , k . This graph is endowed with anested ear decomposition with k + 1 even ears. According to Theorem 4.4,reg G = k + k +1 − ( q −
2) = ( k − q − . k k − k − k +1 3 k +24 k Figure 15.
On the other hand, as any set of independent vertices must have at most twoelements in the k cycles of length 4 created by the addition of the k ears and thevertex sets of these cycles cover V G , we deduce that α ( G ) is 2 k . In conclusion, thisexample shows that, indeed,reg G − ( α ( G ) − q −
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