Toric ideals associated with gap-free graphs
In this paper we prove that every toric ideal associated with a gap-free graph has a squarefree lexicographic initial ideal. Moreover, in the particular case when the complementary graph of is chordal (i.e. when the edge ideal of has a linear resolution), we show that there exists a reduced Gröbner basis of the toric ideal of such that all the monomials in the support of are squarefree. Finally, we show (using work by Herzog and Hibi) that if is a monomial ideal generated in degree , then has a linear resolution if and only if all powers of have linear quotients, thus extending a result by Herzog, Hibi and Zheng.
Key words and phrases:Gap-free graph, Gröbner basis, toric ideal, linear resolution, linear quotients
2010 Mathematics Subject Classification:13P10, 05E40
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Algebraic objects depending on combinatorial data have attracted a lot of interest among both algebraists and combinatorialists: some valuable sources to learn about this research area are the books by Stanley [Stanley], Villarreal [Villarreal], Miller and Sturmfels [MillerSturmfels], and Herzog and Hibi [HerzogHibi]. It is often a challenge to establish relationships between algebraic and combinatorial properties of these objects.
Let be a simple graph and consider its vertices as variables of a polynomial ring over a field . We can associate with each edge of the squarefree monomial of degree 2 obtained by multiplying the variables corresponding to the vertices of the edge. With this correspondence in mind, we can now introduce some algebraic objects associated with the graph :
the edge ideal is the monomial ideal generated by ;
the toric ideal is the kernel of the presentation of the -algebra generated by .
An important result by Fröberg [Fr] gives a combinatorial characterization of those graphs whose edge ideal admits a linear resolution: they are exactly the ones whose complementary graph is chordal. Another strong connection between the realms of commutative algebra and combinatorics is the one which links initial ideals of the toric ideal to triangulations of the edge polytope of , see Sturmfels’s book [Sturmfels] and the recent article by Haase, Paffenholz, Piechnik and Santos [HPPS]. Furthermore, Gröbner bases of have been studied among others by Ohsugi and Hibi [OHToric] and Tatakis and Thoma [TTUniversal]. A necessary condition for to have a squarefree initial ideal is the normality of , which was characterized combinatorially by Ohsugi and Hibi [OHPolytopes] and Simis, Vasconcelos and Villarreal [SVV]. Normality, though, is not sufficient: Ohsugi and Hibi [OHStar] gave an example of a graph such that is normal but all possible initial ideals of are not squarefree.
An interesting class of graphs is the one consisting of the so-called gap-free graphs (following Dao, Huneke and Schweig’s notation in [DHS]), i.e. graphs such that any two edges with no vertices in common are linked by at least one edge. Unfortunately, these graphs do not have a standard name in the literature. Just to name a few possibilities:
graph theorists refer to gap-free graphs as “2-free graphs” and so do Hibi, Nishiyama, Ohsugi and Shikama in [HNOS];
Nevo and Peeva call them “-free graphs” in [Nevo] and [NevoPeeva];
Ohsugi and Hibi use the phrase “graphs whose complement is weakly chordal” in [OHIndispensable];
Corso and Nagel call bipartite gap-free graphs “Ferrers graphs” in [CorsoNagel].
The main goal of this paper is to prove that the toric ideal has a squarefree lexicographic initial ideal, provided the graph is gap-free (Theorem 3.9): moreover, the corresponding reduced Gröbner basis consists of circuits. In the particular case when has a linear resolution (Theorem 3.6) we are actually able to prove that the reduced Gröbner basis we describe consists of circuits such that all monomials (both leading and trailing) in the support of are squarefree, thus extending a result of Ohsugi and Hibi [OHCompressed] on multipartite complete graphs.
In [HHZ] Herzog, Hibi and Zheng proved that the following conditions are equivalent:
has a linear resolution;
has linear quotients;
has a linear resolution for all .
It is quite natural to ask (see for instance the article by Hoefel and Whieldon [HW]) whether these conditions are in turn equivalent to the fact that
has linear quotients for all .
In Theorem 2.6 we prove that this is indeed the case, as can be deduced from results in [HerzogHibi]. Note that all the equivalences between conditions (a), (b), (c), (d) above hold more generally for monomial ideals generated in degree 2 which are not necessarily squarefree.
The computer algebra system CoCoA [CoCoA] gave us the chance of performing computations which helped us to produce conjectures about the behaviour of the objects studied.
2. Notation and known facts
First of all, let us fix some notation. will always be a field and a simple graph with vertices and edges . We can associate to each edge the degree 2 monomial (called edge monomial) and hence we can consider the edge ideal and the subalgebra . In the following we will denote by the toric ideal associated with , i.e. the kernel of the surjection
Since the algebraic objects we defined are not influenced by isolated vertices of , we will always assume without loss of generality that does not have any isolated vertex. We will now introduce some terminology and state some well-known results about toric ideals of graphs: for reference, see for instance [HerzogHibi, Section 10.1].
A collection of (maybe repeated) consecutive edges
(also denoted by ) is called a walk of . If , the walk is closed. If is even (respectively odd), the walk is an even (respectively odd) walk. A path is a walk having all distinct vertices; a cycle is the closed walk most similar to a path, i.e. such that vertices are all distinct. A bow-tie is a graph consisting of two vertex-disjoint odd cycles joined by a single path. Given a walk , we will denote by the subgraph of whose vertices and edges are exactly the ones appearing in . If no confusion occurs, we will often write walks in more compact ways, such as by decomposing them into smaller walks. If is a walk, denotes the walk obtained from by reversing the order of the edges.
If is an even closed walk, one can associate with a binomial in the following way:
where, if , by we mean the variable which is mapped to by the standard surjection. A subwalk of is an even closed walk such that all even edges of are also even edges of and all odd edges of are also odd edges of . An even closed walk is called primitive if it does not have any proper subwalk. The set of binomials corresponding to primitive walks of a graph coincides with the so-called Graver basis of (see for instance [Sturmfels]) and is denoted by .
Note that, given a primitive walk , one can paint – using two colours – the edges of so that those appearing in an even position in are assigned the same colour and those appearing in an odd position are assigned the other one. If an edge were assigned both colours, then the walk would not be primitive: deleting inside both monomials one instance of the variable corresponding to that edge, one could construct a proper subwalk of .
The support of a binomial is the union of the supports of the monomials and , that is to say the variables that appear in and . A binomial is called a circuit if it is irreducible and has minimal support, i.e. there does not exist such that . The set of circuits of is denoted by .
Let be an ideal of . A Gröbner basis of with respect to a term order is called reduced if every element of is monic, the leading terms of minimally generate and no trailing term of lies in . Such a basis is unique and is denoted by . Generally speaking, changing the term order yields a different reduced Gröbner basis: we will denote by the universal Gröbner basis of , i.e. the union of all reduced Gröbner bases of .
Proposition 2.2 ([Sturmfels, Proposition 4.11]).
One has that .
The second inclusion of Proposition 2.2 means that every reduced Gröbner basis of consists of binomials coming from primitive walks of . Consider the set of monomials (both leading and trailing) in such a basis: if they are all squarefree, we will say that is doubly squarefree.
Complete characterizations of both (Villarreal [VillarrealCircuits]) and (Tatakis and Thoma [TTUniversal]) are known. We recall the characterization of (using the phrasing in Ohsugi and Hibi’s article [OHCircuits]) as a reference.
A binomial is a circuit of if and only if , where is one of the following even closed walks:
an even cycle;
where and are odd cycles with exactly one common vertex;
where and are vertex-disjoint odd cycles and is a path running from a vertex of to a vertex of .
Let be a graded ideal generated in degree .
If the minimal free resolution of as an -module is linear until the -th step, i.e. for all , , we say that is k-step linear.
If is -step linear for every , we say has a linear resolution.
If is minimally generated by and for every one has that is generated by elements of degree 1, then is called a linear quotient ordering and is said to have linear quotients.
If for some graph and has one of the properties above, we say that has that property.
Proposition 2.5 ([HerzogHibi, Proposition 8.2.1]).
Let be a graded ideal generated in degree . Then
We now recall an important result by Herzog, Hibi and Zheng ([HHZ]) about the connection between linear quotients and linear resolution in the case when is a monomial ideal generated in degree 2. Condition (d) below did not appear in the original paper: its equivalence to other conditions, though, can be obtained quickly using results in [HerzogHibi].
Let be a monomial ideal generated in degree 2. Then the following conditions are equivalent:
has a linear resolution;
has linear quotients;
has a linear resolution for all ;
has linear quotients for all .
The implications and are obvious, while and follow from Proposition 2.5. It is then enough to prove that , but this follows at once from [HerzogHibi, Theorems 10.1.9 and 10.2.5] (since the lexicographic order introduced in Theorem 10.2.5 is of the kind appearing in Theorem 10.1.9). ∎
Theorem 10.2.5 and the proof of Theorem 10.1.9 in [HerzogHibi] (or, as an alternative, just the proof of the implication in Theorem 10.2.6) tell us also that, if is a monomial ideal of degree 2 having a linear resolution and is a minimal set of monomial generators for , then there exists a permutation of such that is a linear quotient ordering for . As a consequence, if is the edge ideal of some graph having a linear resolution, there exists a way of ordering the edge monomials so that they form a linear quotient ordering.
We thank Aldo Conca for pointing out the following result:
Let be distinct homogeneous elements of degree in which are minimal generators for the ideal . The following conditions are equivalent:
is a linear quotient ordering;
the ideal is 1-step linear for all .
Let us prove that . Let . If is a linear quotient ordering, than is too and hence, by Proposition 2.5, the ideal has a linear resolution; in particular, it is 1-step linear.
To prove that , let . Consider the exact sequence
where is the map which sends to for all . Then, by hypothesis, is generated in degree 1. Since is isomorphic to the -th projection of , we are done. ∎
In what follows, we will denote by the complementary graph of , i.e. the graph which has the same vertex set of and whose edges are exactly the non-edges of .
The next result by Eisenbud, Green, Hulek and Popescu proves that, in our context, the algebraic concept of -step linearity can be characterized in a purely combinatorial manner.
Proposition 2.9 ([Eghp, Theorem 2.1]).
Let be a graph and let . The following conditions are equivalent:
is -step linear;
does not contain any induced cycle of length for any .
As a corollary, we recover the important result by Fröberg characterizing combinatorially graphs with a linear resolution.
Corollary 2.10 ([Fr]).
Let be a graph. Then has a linear resolution if and only if is chordal, i.e. does not contain any induced cycle of length greater than or equal to .
Following the notation in [DHS], we will call a graph gap-free if for any , in (where are all distinct) there exist such that . In other words, in a gap-free graph any two edges with no vertices in common are linked by at least a bridge.
It is easy to see that is gap-free if and only if does not contain any induced cycle of length 4. It then follows from Proposition 2.9 that is gap-free if and only if is 1-step linear.
The following theorem holds more generally for affine semigroup algebras.
Let be a graph.
(Hochster [Hochster]) If is normal, then it is Cohen-Macaulay.
(Sturmfels [Sturmfels, Proposition 13.15]) If admits a squarefree initial ideal with respect to some term order , then is normal (and hence Cohen-Macaulay).
The problem of normality of graph algebras (and, as a consequence, of edge ideals, see [SVV, Corollary 2.8]) was addressed and completely solved by Ohsugi and Hibi [OHPolytopes] and Simis, Vasconcelos and Villarreal [SVV]. One of the main results they found is the following:
A connected graph is such that is normal if and only if satisfies the odd cycle condition, i.e. for every couple of disjoint minimal odd cycles in there exists an edge linking and .
Ohsugi and Hibi [OHStar] also found an example of a graph such that is normal but is not squarefree for every choice of , hence the condition in Theorem 2.12.2 is sufficient but not necessary.
There is a strong connection between squarefree initial ideals of and unimodular regular triangulations of the edge polytope of . To get more information about this topic, see [Sturmfels] and the recent work [HPPS], in particular Section 2.4.
We start by stating a result about the shape of primitive walks. This is a modification of [OHToric, Lemma 2.1]: note that primitive walks were completely characterized by Reyes, Tatakis and Thoma in [RTT, Theorem 3.1] and by Ogawa, Hara and Takemura in [OHT, Theorem 1]. In the rest of the paper we will often talk of primitive walks of type (i), (ii), (iii) referring to the classification below.
Let be a primitive walk. Then is one of these:
an even cycle;
where and are odd cycles with exactly one common vertex;
where the ’s are paths of length greater than or equal to one and the ’s are odd cycles such that and are vertex-disjoint for every .
Let be a primitive walk neither of type (i) nor (ii). Since is primitive, there exists a cycle inside (otherwise where is a path and hence all edges of would appear both in odd and even position in , thus violating the primitivity); moreover, since is not of type (i), has to be odd. Let ; then
Let be the least integer such that coincides with one of the vertices in .
Suppose where . If and , we get that the edge is both an even and an odd edge of (contradiction). In all other cases, paint the edges appearing in red and black alternately and note that, since , there are both a red and a black edge of starting from . Then exactly one of and is an even closed subwalk, thus violating the primitivity of . This gives us a contradiction.
Suppose where (since has no loops, ). Note that one actually has that , since would imply that the edge is both an even and an odd edge of (contradiction). Therefore there exists a cycle disjoint from by construction. Since is primitive, must be odd; moreover, since is not of type (ii), one has that . This means that we have found a path linking the odd cycles and . We can now repeat the whole procedure starting from the cycle to find a path and an odd cycle disjoint from and so on, hence proving the claim in a finite number of steps. ∎
The referee noted that an alternative proof of Lemma 3.1 may be given using [OHT, Theorem 1].
Note that, by Proposition 2.3, all binomials corresponding to primitive walks of type (i) and (ii) are circuits.
Let be a graph with edges and let be a term ordering on . With a slight abuse of notation, we will often say that instead of (where ). Moreover, if is a subgraph of and is lexicographic, we will say that is the leading edge of with respect to if for every .
Next we introduce the main technical lemma of the paper. Note that, when dealing with the vertices of a cycle, for the sake of simplicity we will often write instead of .
Let be a primitive closed walk of of type (iii) and let be a lexicographic term order on . Let be the leading edge of with respect to : by Lemma 3.1, lies into a bow-tie . Let , and let and be the starting and ending vertices of the path . Suppose one of the following two conditions holds:
and there exist such that , , ;
and there exists such that at least one between and is an edge of (call it ) such that and .
First of all, by Remark 2.1 the primitivity of the walk allows us to paint the edges of red and black so that no two edges consecutive in are painted the same colour. We can assume without loss of generality that the edge is black.
Paint red. We can suppose without loss of generality that : hence, exactly one of and is black. This means that exactly one of the two paths going from to along has its first edge painted black: let be this path. We now need to define a path going from to .
If , exactly one of and is black. Applying the same reasoning as before, let be the path going from to along having its last edge painted black.
If and the last edge of is red, let (in other words, the whole cycle ); if the last edge of is black, let be the empty path in .
Let . By construction, is an even closed walk, since its edges are alternately red and black and the first and the last one have different colours. Moreover, it is easy to check that is primitive either of type (ii) (when and the last edge of is red) or of type (i) (in all other cases); hence, . Finally, since is a lexicographic term order, to get who the leading monomial of is we just have to identify the leading edge of : since by hypothesis and the rest of the edges of are edges of , we get that the leading monomial of is the one formed by black edges. Since the black edges of all lie in , we have that divides . Since , we have that .
Paint red and define in the same way as in part (a). Let be defined the following way:
if , let (if , is the empty path);
if , let (if , is the empty path).
Reasoning the same way as in part (a), we get that is an even closed walk; moreover, it can be easily checked that is primitive either of type (ii) (when belongs to and the last edge of is red or when belongs to , with no restrictions on the colour of the last edge of ) or type (i) (in all other cases), hence . For the same reasons as in part (a), we get that . ∎
Let be a graph with linear resolution and let be an ordering of the edges of such that is a linear quotient ordering for (such an ordering exists by Remark 2.7). Let be the lexicographic order on such that . Then the reduced Gröbner basis of with respect to is doubly squarefree.
By Proposition 2.8, the linear quotient property is equivalent to asking that each subgraph is 1-step linear, that is to say gap-free by Remark 2.11. Let be a primitive walk such that at least one of the two monomials of is not squarefree. This implies that is primitive of type (iii). Hence, by Lemma 3.1, we know that the leading edge of lies into a bow-tie . Let be the subgraph of obtained by considering all the edges such that . This means that ; hence, is gap-free. Using the notation of Lemma 3.5, we have to consider two different cases.
If , consider the edges and . Since is gap-free, there exists an edge which links the edges we are considering and is such that . By applying Lemma 3.5.(a), we get that .
If , consider the edge . Reasoning as before, we discover the existence of an edge linking these two edges and having the property that : hence, by applying Lemma 3.5.(b), we get that .
This ends the proof. ∎
As a corollary we recover a result by Ohsugi and Hibi [OHCompressed] about complete multipartite graphs:
Corollary 3.7 ([OHCompressed]).
If is a complete multipartite graph, then there exists a doubly squarefree Gröbner basis of .
The complementary graph of a complete multipartite graph is a disjoint union of cliques and hence is chordal. Applying Theorem 3.6 yields the thesis. ∎
In Theorem 3.6 we actually proved that does not contain any binomials corresponding to primitive walks of type (iii). This means in particular that consists entirely of circuits (and hence is generated by circuits, as one could have already noticed applying Theorem 2.6 in Ohsugi and Hibi’s article [OHToric]).
Let be a gap-free graph and order its edges the following way: if and only if , where is an arbitrary graded reverse lexicographic order on . Rename the edges so that . Let be the lexicographic order on such that . Then is generated by squarefree elements.
Let be a primitive walk of such that is not squarefree and let be the leading edge of with respect to . Then, since has to be of type (iii), by Lemma 3.1 there exists a bow-tie containing . We will use the notation of Lemma 3.5 to denote the edges of this bow-tie.
If , then no edges of have vertices in common with . In the following we will say that a vertex satisfies condition if
Note that, by definition of and , if an edge shares no vertices with , then at least one of and must satisfy condition . Since no edges of share vertices with , any pair of consecutive vertices in must include a vertex satisfying condition : since is odd, by pigeonhole principle we get that there exists an edge of whose vertices both satisfy condition .
Since is gap-free, there exists linking and : moreover, since both vertices of satisfy condition , one has that . If then, by Lemma 3.5.(b), we get that . If , then and we have to consider two different cases.
If is made of an even number of edges, then is a primitive walk of type (ii) such that divides . Hence .
If is made of an odd number of edges, then consider . By Proposition 2.3, is a circuit and hence is a primitive walk. Since is squarefree and divides , we get that .
If , we have to discuss two different situations.
If is made of more than one edge, then at least one of the cycles and has no vertices in common with (let it be without loss of generality). Then, applying the same pigeonhole reasoning used in the previous case, we discover the existence of an edge of whose vertices both satisfy condition . Since is gap-free, there exists linking and . Since by construction, applying Lemma 3.5.(a) we get that .
The last case standing is the one where . Let , . If there exist two consecutive vertices belonging to either or and satisfying condition , then we can apply Lemma 3.5.(a) to infer that . Suppose otherwise. Then condition is satisfied alternately: to be more precise, we have that the vertices of (or ) satisfying condition are either the ones with odd index or the ones with even index. We can suppose without loss of generality that are the vertices in satisfying condition . Consider the edges and . Since is gap-free, these edges are surely linked by some edge : if one of and belongs to we have that and hence, by Lemma 3.5.(a), we can conclude that . What happens if ? If we are done for the same reason as before. Suppose . Then, by definition of , at least one of and (call it ) must be such that and . Since is the leading edge of , though, one has that and , hence and . This gives us a contradiction. ∎
The proof of Theorem 3.9 shows also that consists of circuits and hence (as we already knew by [OHToric, Theorem 2.6]) is generated by circuits. To see this, replace the hypothesis “ primitive walk such that is not squarefree” with “ primitive walk of type (iii)” and note that the only primitive walks of type (iii) that may appear in are bow-ties (more precisely, just those with a connecting path of length one). Since binomials associated with bow-ties are circuits by Proposition 2.3, we are done.
In general, the construction appearing in Theorem 3.9 does not necessarily yield a doubly squarefree reduced Gröbner basis of . For instance, consider the gap-free graph with vertices and the following edges:
Note that the edges are ordered from the biggest to the smallest in a reverse lexicographic way according to the vertex order 1 > 2 > 3 > 4 > 5 > 6 (in the sense explained in the claim of Theorem 3.9). Let be the lexicographic order on such that . Then CoCoA computations yield
The author would like to thank his advisor Aldo Conca, Matteo Varbaro and the referee for their valuable comments and suggestions.