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Abstract

In combinatorial commutative algebra and algebraic statistics many toric ideals are constructed from graphs. Keeping the categorical structure of graphs in mind we give previous results a more functorial context and generalize them by introducing the ideals of graph homomorphisms. For this new class of ideals we investigate how the topology of the graphs influence the algebraic properties. We describe explicit Gröbner bases for several classes, generalizing results by Hibi, Sturmfels and Sullivant. One of our main tools is the toric fiber product, and we employ results by Engström, Kahle and Sullivant. The lattice polytopes defined by our ideals include important classes in optimization theory, as the stable set polytopes.

Graph homomorphisms, toric ideals, Gröbner bases, algebraic statistics, structural graph theory

Ideals of Graph Homomorphisms]Ideals of Graph Homomorphisms

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Primary 05C60; Secondary 68W30, 13P25, 13P10, 62H17

1 Introduction

In this paper we introduce the ideals of graph homomorphisms. They are natural generalizations of toric ideals studied in particular in combinatorial commutative algebra and algebraic statistics. The lattice polytopes associated to them are important in optimization theory, and we can derive results on graph colorings with these ideals. Many toric ideals in the literature are defined from graphs, but usually the categorical structure is lost in the translation. Defining the objects from graph homomorphisms provide functorial constructions for free, as in homological algebra.

1.1 A short overview of the paper

For every pair of graphs and the graph homomorphisms from to defines a toric ideal . In Section 4 we give a proper definition of ideals of graph homomorphisms . We give examples and explain how they relate to previously studied toric ideals, in particular from algebraic statistics. Some basic properties are proved, with focus on how modifications of the graphs and change the ideal of graph homomorphisms from to .

The toric fiber product introduced by Sullivant [37] and further developed by Engström, Kahle, and Sullivant [13] is explained in the context of ideals of graph homomorphisms in Section 5. If the intersection of two graphs and is sufficiently well-behaved with regard to a target graph , this allows us to lift algebraic properties and bases of and to . We apply this to understand for in some graph classes.

In Section 6 we review results on normality, and how to use the toric fiber product of the previous section to lift normality from particular graphs to complete classes. For example, we show that if the semigroup associated to is normal, then so are the ones associated to when is a maximal outerplanar graph.

The independent sets of a graph are indexed by the graph homomorphism from to the graph with two adjacent vertices and one loop. In Section 7 we study the ideals . First we give a convenient multigrading that is crucial for later proofs. For many families of toric ideals in algebraic statistics it is known, or conjectured, that the largest degree of an element in a minimal Markov basis is even. We show by explicit constructions that also odd degrees appear for . Then we derive a quadratic square-free Gröbner basis for when is a bipartite graph, and this shows that they are normal and Cohen-Macaulay.

Following Section 8, we extend our results from bipartite graphs to graphs that become bipartite after the removal of some vertex. This is much more technically challenging.

Our toric ideals define lattice polytopes . In Section 9 we first study how modifications of gives faces of . Then we show how one of the most important classes of polytopes in optimization theory, the stable set polytopes [30], appear naturally as isomorphic to some of our polytopes.

In Section 10 we show that Hibi’s algebras with a straightening law from distributive lattices [23] is isomorphic to a graded part of some particular ideals of graph homomorphisms. Hibi’s results on normality, Cohen-Macaulayness, and Koszulness, follows right off from our much larger class.

The ideal of graph homomorphisms whose target graph is a complete graph, , is an algebraic structure on the -colorings of a graph . In Section 11 we show how the ideals can be used to give structural information about graph colorings.

But we start off by introducing some notation from toric geometry and algebraic statistics in Section 2, and a short overview of the category of graphs in Section 3.

2 Toric geometry in algebraic statistics

The toric ideals studied in this paper are closely connected to those in algebraic statistics. While the methods from any textbook on combinatorial commutative algebra, like Miller and Sturmfels [32], is enough to parse most algebraic statements of this paper, we want to point out some notions and particularities of algebraic statistics. For a nice introduction to this area we recommend the lectures on algebraic statistics by Drton, Sturmfels and Sullivant [11].

We fix a field throughout the paper. Two equivalent ways to define a toric ideals are used: For a matrix and a polynomial ring the toric ideal is generated by the binomials for which . Alternatively we could have defined as the kernel of the map from to defined by . This toric ideal cuts out a toric variety denoted or .

For each monomial in , the fiber of is the set of monomials in mapped to by . For any monomials and in the same fiber, there is a binomial in the toric ideal and a monomial in such that . If is a Markov basis (that is, a generating set) of , then there is a sequence of monomials

in the fiber such that for some binomial generator in and monomial in , for all . The step from to in the sequence is referred to as a Markov step or a Markov move.

There is a graph structure on the fiber of the monomial . This graph has the monomials of the fiber as vertices, and they are adjacent if there is a Markov step between them. A set of binomials is a basis if and only if every fiber graph is connected. We often view the Markov steps as having a direction imposed by the basis. If a basis is constructed with an order for each binomial , then this imposes an order on each Markov steps, and turns the fiber graphs directed. A sink in a directed graph has all edges directed towards it, and in a directed acyclic graph there are no cycles with the edges directed in consecutive order. If all fiber graphs of are connected, directed acyclic, and have a unique sink, then is a Gröbner basis.

The degree of a basis is the maximal degree of a binomial in it. The Markov width of a toric ideal , denoted , is the minimal degree of a basis of . The Markov width is an important invariant of and a good complexity measure. When the toric ideals are given by graphs, it is a central question how the topological and structural properties of the graphs are reflected in the Markov width [13].

3 The category of graphs

The reader is invited to recall basic graph theory from Diestel [9]. A loop is an edge attached to only one vertex. Most our graphs are simple or with loops, but never with multiple or weighted edges. We get the graph from by attaching loops to all its vertices. Although we sometimes have loops, the complement of a graph without loops, is the ordinary complement without loops. The symmetric difference of two sets and contains the elements that are in exactly one of and . For an integer the set is . The neighborhood is the set of vertices adjacent to in a graph (excluding loops), and for any set of vertices. The induced subgraph of on vertex set is denoted . A particular case of this, is for any edge of , the graph is the subgraph of only containing the edge . The independence target graph in Figure 1 will appear frequently in later parts of the paper.

Figure 1: The graph
Definition 3.1.

A graph homomorphism from a graph to a graph is a function from the vertex set of to the vertex set of that induces a function from the edge set of to the edge set of . A more formal way of stating it is that

satisfy

The set of graph homomorphisms from to is denoted . A graph isomorphism from to is a graph homomorphism from to such that is a bijection between and , and the inverse of is a graph homomorphism from to . If such a map exists then and are said to be isomorphic and can be considered to be the same.

The following trivial facts are useful, and they capture the first aspect of why defining ideals from sets of graph homomorphisms might be a structural theory.

Lemma 3.2.

Let and be graphs. If and then .

Proof.

The map induces to edge sets . ∎

The graph is a subgraph of , denoted , if and

Lemma 3.3.

Let be graphs. If and then

Proof.

Take any , and set for . Then use Lemma 3.2 to conclude that . ∎

Lemma 3.4.

Let be graphs. If and then

Proof.

Let be the inclusion map in , and take any . Then by Lemma 3.2, . ∎

If is a graph and a subset of , then any map can be restricted to a map . In particular, a graph homomorphisms restricts to a graph homomorphism . For future reference we state this as a trivial lemma without proof.

Lemma 3.5.

If are graphs, , and , then

For more about graph homomorphisms, see the textbook by Hell and Nešetřil [20].

4 Ideals of Graph Homomorphisms

In this section we define and prove the basic properties of the ideals of graph homomorphisms. These ideals are toric ideals defined as kernels, and we first define rings and a map.

Definition 4.1.

For any graphs and , the ring of graph homomorphisms from to is the polynomial ring

and the ring of edge maps from to is the polynomial ring

For every the domain of the graph homomorphism is a graph consisting of one edge and its vertices. By Lemma 3.5 the following edge separator map is well defined and a ring homomorphism.

Definition 4.2.

For graphs and the edge separator map

is defined by

Now we are ready to define the object of our study.

Definition 4.3.

For graphs and the ideal of graph homomorphisms from to , , is the kernel of the edge separator map .

The corresponding toric variety is denoted , and the lattice polytope .

Example.

The hierarchical model with variables taking discrete values modeled on a graph , is a common statistical model that is studied in algebraic statistics. Recall that is the complete graph on vertices with loops on all vertices. The hierarchical model is the special case of ideals of graph homomorphisms . Many properties and examples of these models have been studied. Develin and Sullivant [7] studied the Markov width of binary graph models and constructed Markov bases of degree four for the binary graph models when the graphs are cycles and complete bipartite graphs . Hoşten and Sullivant [25] gave Gröbner basis for the binary graph model when the graph is a cycle and found a complete facet description of the underlying polytope. Another common model in statistics is the graphical models, they are associated to the ideals if does not contain a triangle. Hierarchical models are defined in terms of simplicial complexes, the case when the complex is the clique complex of a graph is called graphical. Geiger, Meek and Sturmfels [18] found conditions for when a statistical model is graphical. Kahle [26] found a neighborliness property of the underlying polytope for hierarchical models. Likelihood estimation for hierarchical models is studied for example in [8, 10, 14].

Example.

An independent set of a graph is a set of non-adjacent vertices. Another name for independents sets are stable sets, and most of the important graph theoretic concepts and problems can be stated as properties of them. Recall that the graph on two vertices with one edge and one loop is denoted . Every independent set of a graph can be described as a graph homomorphism from into where the independent set is the pre-image of the vertex without a loop. The ideal of graph homomorphisms , or the ideal of independent sets, is an important special case that we will return to in Section 7. The polytope associated to is isomorphic to the stable set polytope, which is important in optimization theory.

Example.

A graph coloring is an assignment of colors to the vertices of a graph with no adjacent vertices getting the same color. The minimal number of colors, the chromatic number, is an important invariant of a graph. An upper bound for the chromatic number can easily be achieved by giving an explicit coloring, but to prove that a certain number of colors is indeed needed, is much more difficult. There are many simplistic ways to associate algebraic structures to graphs in attacking this problem, but the only successful ones so far makes heavy use of the underlying category of graphs and their homomorphisms [3]. A graph coloring of a graph with vertices is nothing but a graph homomorphism from to the complete graph . In Section 11 we will show how ideals of graph homomorphisms can be used in the study of graph colorings.

Example.

Let be the path on the four vertices 1,2,3,4; and the path on the three vertices 1,2,3; as in Figure 2. The variable of the ring of graph homomorphisms corresponding to a graph homomorphism is called . The variables are and the ideal of graph homomorphisms is generated by and

Figure 2: The domain and target of the graph homomorphisms defining .

If have isolated vertices then contains lots of uninteresting quadratic binomials. Most graphs we study lack isolated vertices, but we don’t restrict to that case. If you change the source or target for a set of graph homomorphisms, it is also reflected in their rings.

Lemma 4.4.

Let be graphs. If and then .

Proof.

Use Lemma 3.3 and Definition 4.1. ∎

Lemma 4.5.

Let be graphs. If and then .

Proof.

Use Lemma 3.4 and Definition 4.1. ∎

Ordinarily we don’t want to expand the target of our graph homomorphisms as in Lemma 4.4, but to move around in subrings where the target is reduced. This is handled in Lemma 4.6.

Lemma 4.6.

Let be graphs with and ; and let be monomials in . If then either both or both .

Proof.

By symmetry of and , we only have to prove that if then . Assume that since for some graph homomorphism the variable divides . This is certified by an edge of mapped to in . The images of and under are the same, so there is a graph homomorphism such that divides and sends to . This shows that and hence is not in . ∎

Theorem 4.7.

Let be graphs with and . If is a basis of , then is a basis of .

Proof.

If and are monomials in and , then there are monomials such that

and each binomial equals some where is a monomial in and is a binomial in . We want to show that each is in to prove that . To do this we find that each is in .

We assumed that , so in particular . By Lemma 4.6 then also . Repeating the same argument, gives that all , and there differences , are in . ∎

Corollary 4.8.

If are graphs with and then the Markov widths are related by

Proof.

Let be a degree basis of . Restricting to gives a basis according to Theorem 4.7, and that one is at most of the same degree as . ∎

5 Gluing together graphs

In structural graph theory it is studied how graph classes either can be defined by forbidden minors, or by being glued together from simple starting graphs [33]. In algebraic statistics, when ideals are formed from graphs, one can ask if there is an operation on the level of ideals corresponding to gluing the graphs. The first algebraic result in this direction, collecting several scattered results and giving them a theoretical foundation, was obtained by Sullivant [37] when he defined the toric fiber product and showed how to make use of it in the codimension zero case. In codimension one the first result was proved by Engström [12] and it was used to prove that cut ideals of -minor free graphs are generated by quadratic square-free binomials, as conjectured by Sullivant and Sturmfels [36]. The first systematic treatment of higher codimensions, with a clear connection to structural graph theory, was recently done by Engström, Kahle, and Sullivant [13]. In this section we use the toric fiber product to find generators of ideals of graph homomorphisms.

The integer matrix in the definition of a toric ideal can also be regarded as a configuration of integer vectors. For two vector configurations and we get toric ideals in and defined by and Assume that there is a vector configuration and linear maps satisfying and for all the vectors. Their toric fiber product is the toric ideal

in where

Proposition 5.1.

Let and be graphs. If is an induced subgraph of both and then

Proof.

Let be the vector configuration defining the toric ideal . Any graph homomorphism restricts to a graph homomorphism This gives the linear –maps from the vector configurations defining and to . ∎

When the subscript of is clear, as it almost always is in our applications of the toric fiber product, then we drop it from the notation. The easiest toric fiber products to work with are when the vectors in are linearly independent, because then there is a procedure to get the basis of the product from the bases of the factors. We now describe this procedure of Sullivant [37] in the context of ideals of graph homomorphism.

Proposition 5.2.

Let be a generating set of for and let be the vector configuration defining . If is an induced subgraph of both and , and the vectors of are linearly independent, then

is a generating set of , where is the set

is the set

and is the set

Proof.

This is a direct application of Corollary 14 in [37], where in a general context Quad is defined in Proposition 10 and Lift is defined in Definition 11. This setup is also discussed in [13] in a more general context. ∎

Our main use of the previous proposition is a natural extension of the similar results for hierarchical models.

Lemma 5.3.

Let and be two graphs whose intersection is one of or ; and let be a graph. Then

Proof.

By Proposition 5.1 the ideal of graph homomorphisms is the toric fiber product . The vector configurations defining the toric ideals are linearly independent if is one of . We apply Proposition 5.2 to bound the Markov width. Let be a generating set of with binomials of degree at most for By construction in Proposition 5.2 the binomials in are of degree at most , the binomials in Quad are quadrics, and hence since generates . ∎

Theorem 5.4.

If is a forest then is generated by square-free quadratic binomials.

Proof.

If is a vertex this is true. We defer the case of that has several components to the end and assume that is a tree. The proof is by induction on the number of edges. If is an edge then is trivial. Otherwise cover by two trees and that both have at least one edge such that they intersect in a vertex. By induction both and are generated by quadrics, and then so is their union by Lemma 5.3. That they are square-free follows from that square-free binomials lifts to square-free, and that all binomials from Quad are square-free, in Proposition 5.2

If is not a tree but a forest, then the same argument but gluing over empty sets apply. ∎

An outerplanar graph is a graph that can be drawn in the plane with straight edges and with its vertices on a circle without any edges crossing each other. A maximal outerplanar graph is thus a triangulation of an –gon.

Theorem 5.5.

If is a maximal outerplanar graph on at least three vertices, then

Proof.

The proof is by induction on the number of triangles in . The statement is clearly true if is a triangle. If has more than one triangle, then there is a way to decompose into graphs and such that both of them are maximal outerplanar graphs with at least one triangle, and their intersection is an edge. By an application of Lemma 5.3 we are done. ∎

Example.

The ideal of graph homomorphisms of four-colorings of a maximal outerplanar graph , , is generated by binomials of degree 2 and 12. To see why this is true we not only need Theorem 5.5, but also the explicit description in Proposition 5.2. Using the 4ti2 software [1] we computed that the toric ideal is generated by the degree 12 binomial

The binomial can be described using a permutation representation of the alternating group on four elements. When we glue together two maximal outerplanar graphs, any binomial of degree 12 will lift to a binomial of degree 12. The quadratics will lift to quadratics, and the Quad moves will only give quadratics.

Propostion 5.2 is a Corollary of a Theorem about Gröbner bases by Sullivant [37]. For future reference we state this theorem in the special case of ideals of graph homomorphisms. There is another useful type of partial order on monomials called a weight order: Let be a vector of weights. The weight order on the monomials in the variables is defined by if . A Gröbner basis of an ideal with respect to a weight order is a finite generating set of with the property that the initial monomials of generate the initial ideal of .

Let be a homomorphism between polynomial rings such that sends each variable to a monomial. A weight vector for the image of induces weight vector on the domain such that the weight of a monomial is the weight of the image of the monomial.

Let and be graphs such that their edge sets agree on their intersection and let . Define a ring homomorphism from to by

Proposition 5.6.

Let be a Gröbner basis of with respect to for and let be the vector configuration defining . Assume that is a Gröbner basis with respect to . If the vectors of are linearly independent, then

is a Gröbner basis of with respect to for sufficiently small .

Proof.

This is Theorem 13 in [37] applied to ideals of graph homomorphisms. ∎

6 Normality and related algebraic properties

In this section we very briefly survey some of the typical algebraical properties that are consequences of a good combinatorial understanding of generating sets of toric ideals. For more discussions of these topics we refer to Fröberg for Koszul algebras [15], Hochster for normal semigroups [24], and Bruns and Herzog for Cohen-Macaulay rings [4].

If is a toric ideal in a polynomial ring over a field , then is isomorphic to a semigroup ring where is a semigroup [6]. In Chapter 13 of Sturmfels textbook on Gröbner bases and polytopes [35] it is proved that if a toric ideal has a square-free Gröbner basis, then its associated semigroup is normal. It is a theorem of Hochster [24] that if is a homogenous toric ideal in whose associated semigroup is normal, then is Cohen-Macaulay. The last two statements are usually bundled up:

Proposition 6.1.

If a homogenous toric ideal in has a squarefree Gröbner basis, then its associated semigroup is normal, and is Cohen-Macaulay.

The following proposition was proved by Anick [2].

Proposition 6.2.

If is an ideal with a quadratic Gröbner basis in a ring , then is Koszul.

Many results about normality in algebraic statistics can be derived from the results of Section 5 in a paper by Engström, Kahle, and Sullivant [13]. We will now explain that method in the context of ideals of graph homomorphisms using the toric fiber product described in the previous section of this paper.

Lemma 6.3.

Let for be ideals whose semigroups are normal, and let be the vector configuration defining . If is an induced subgraph of both and , and the vectors of are linearly independent, then the semigroup associated to is normal.

Using this lemma we can proceed as in Theorem 5.5 to lift results from small graphs to complete classes.

Proposition 6.4.

Let be a graph with normal, then for every maximal outerplanar graph , the ideal is normal.

Proof.

Use the same recursive gluing procedure as in the proof of Theorem 5.5 and lift the property of normality in each step by Lemma 6.3. ∎

In the same spirit, but using the proof of Theorem 5.4 as a template, one can see that is normal whenever is a forest. On the other hand, by an easy slight sharpening of Theorem 5.4, we know that these ideals have quadratic square-free Gröbner bases, and are normal and Cohen-Macaulay by Proposition 6.1.

7 Ideals of graph homomorphisms from independent sets

In this section we study ideals of graph homomorphisms from independent sets. An independent set of a graph can be represented as a graph homomorphism from into the graph by sending all vertices of the independent set onto the vertex without the loop, and the other ones onto the looped vertex. The indeterminate representing the independent set of is denoted .

We now introduce a multigrading on by

for any vertex of . This extends to any monomial by . To determine the kernel of the map we only need the multigrading according to this lemma.

Lemma 7.1.

Let be a graph and let and be monomials in of the same degree. Then the binomial is in if and only if for all vertices of .

Proof.

That the multidegrees of and are equal when their difference is in the kernel is clear, and the proof amounts to showing the other direction. Stated otherwise, we want to show that can be uniquely determined from the multidegree of .

Assume that the total degree of is . An edge can be sent by a graph homomorphism from to in three ways: (1) onto the straight edge with landing on the unlooped vertex, (2) onto the straight edge with landing on the unlooped vertex, and (3) onto the loop. But this is counted by the multidegree. The (1) case occurs times, the (2) case occurs times, and the (3) case occurs times. From this is uniquely determined. ∎

Using Lemma 7.1 it is often easier to argue about the independent sets and the multiset of vertices than about the monomials. Another way of stating the lemma above, is that the difference of two monomials is in the ideal if and only if they give the same multiset of vertices.

7.1 The top graded part

There is another natural grading on the monomials in by the number of vertices in the independent sets. This grading is important since it cuts out ideals that are previously studied. The independence number of a graph is the size of the largest independent set of . Alternatively, could have been defined as the smallest number satisfying for all in One consequence of this inequality, is that if and for all then for all This shows that the following definition makes sense.

Definition 7.2.

The top graded part of is

and the top graded part of is

A toric ideal can be defined in terms of a polytope. This polytope is studied in section 9 but we note here that the top graded part correspond to a face of this polytope. The top graded part of the toric ideal associated to the independent sets of a graph correspond to a face of the polytope.

7.2 Any Markov width is possible

For many toric ideals in algebraic statistics it seems that only even Markov widths are allowed [27]. But this is not the case for ideals of graph homomorphisms from independent sets.

We have performed computations on the ideals for graphs with few vertices. Of all connected graphs with no loops, and eight or fewer vertices, there are with and only four with . All the complete graphs have and the rest have . The graphs with are the graphs with eight vertices depicted in Figure 4. The Markov width is low for all graphs with few vertices, but it does grow and we construct graphs with for any integer in Theorem 7.3.

Example.

The smallest graph with a Markov width larger than two is , the skeleton of a tent. It is two cycles of length where a vertex in one of the cycles is connected with the corresponding vertex in the other cycle. It is drawn in Figure 3. It has a basis containing one element of degree , and it has the quadratic elements .

The graph in the previous example is a special case of a type with arbitrary large Markov width. The next of this type of graph is one of the four on at most eight vertices with Markov width four. It is the complement of a cycle , and it is drawn in Figure 4.

Figure 3: The smallest graph with a Markov width larger than two.
Figure 4: The graphs with at most eight vertices and Markov width four. The rightmost one is the complement of .
Theorem 7.3.

If then

Proof.

Consider the cycle with vertices and edges counting modulo . We prove that the complement of satisfies . Let be the degree binomial

Both of the monomials in has multidegree one for every vertex of , and by Lemma 7.1. The binomial and the quadrics of will form a basis of it.

Say that and are monomials and . We should prove that and can reach each other by Markov moves. The proof is by induction on the degree of . If the degree is two, then by construction of the basis we are done. If the degree of is larger than two, we find Markov moves from to such that and have a common factor, and then we are done by induction on the degree.

So, let and be monomials with no common factors. There are two cases:

  • The monomial (or by symmetry ) contains a factor where is a vertex of .

    The monomial contains or , and without loss of generality we assume the first mentioned. It follows that contains or . If contains then the Markov move from to introduce the common factor . Otherwise contains and the Markov move from to introduce the same common factor.

  • There are no factors in or .

    If contains then contains . And then contains because of that. Proceeding around the cycle we get that contains one of the monomials in and contains the other one. The Markov move using introduces common variables.

In the next section we show that if is bipartite then , and that this is also true if becomes bipartite after removing a vertex. For some 3-partite graphs , but according to Theorem 7.3. We demonstrated the existence of a graph with by a -partite graph, and one could speculate that many parts are forced. It turns out that this is not the case, but it is unclear if is limited for 3-partite graphs.

Theorem 7.4.

For any graph there is a 4-partite graph satisfying

Proof.

We construct