Abstract
The edges of any hypergraph parametrize a monomial algebra called the edge subring of the hypergraph. We study presentation ideals of these edge subrings, and describe their generators in terms of balanced walks on hypergraphs. Our results generalize those for the defining ideals of edge subrings of graphs, which are wellknown in the commutative algebra community, and popular in the algebraic statistics community. One of the motivations for studying toric ideals of hypergraphs comes from algebraic statistics, where generators of the toric ideal give a basis for random walks on fibers of the statistical model specified by the hypergraph. Further, understanding the structure of the generators gives insight into the model geometry.
Introduction
The rich history of the ideal theory of graphs, dating back to the seminal paper [20], shows that fundamental properties of monomial algebras associated to graphs have interesting combinatorial interpretations. The connection inspired a body of work that joins combinatorial and algebraic methods with at least two algebraic objects in focus: monomial edge ideals of graphs, and toric ideals of edge subrings of graphs.
The study of quadratic squarefree monomial ideals corresponding to edges of a graph has generated a lot of interest from the algebraic community. In particular, the comprehensive monograph [23] contains numerous references on the subject. More recent work includes, for example, [3] which describes Betti numbers and primary decompositions for edge ideals of restricted families of graphs. On the other hand, presentation ideals of edge subrings of graphs, which are toric rather than monomial, arise as kernels of monomial maps whose images correspond to the edges of a graph. These toric ideals have been studied in [13, 14, 18, 22, 23].
Several results for the monomial edge ideals have been extended to the case of hypergraphs (see, for example, [6, 9]), where a hypergraph is a generalization of a graph with edges containing more than two vertices. However, toric ideals of edge subrings remain completely unexplored beyond the graph case. Nevertheless, they are of interest in applications, and as blowup algebras, since the edge subring of a uniform hypergraph is isomorphic to the special fiber ring of its edge ideal.
In this work, we extend the theory for these toric ideals from graphs to (uniform) hypergraphs. We study the combinatorial signatures of presentation ideals of monomial subalgebras parametrized by squarefree monomials of degree greater than 2. Their generators, Gröbner and Graver bases are given in terms of monomial walks on hypergraphs, generalizing the constructions that are wellknown for graphs.
One of the motivations for studying toric ideals of hypergraphs comes from algebraic statistics, where generating sets of toric ideals play an important role in testing how well a model fits the given data. The edge subring of a hypergraph corresponds to any exponential family model whose joint probabilities are parametrized by monomials. For more details, see Sect. 3. In fact, edge subrings of graphs are already known in algebraic statistics, most notably in [12, 15, 16, 21]. The first author has used these toric ideals to gain insight into the geometry of a particular random graph model, existence of maximum likelihood estimators, and a better way to generate Markov moves [16, 17]. In particular, this work is a starting point for developing the theory required for the more general algebraic statistical models.
A hypergraph H is duniform if every edge contains d vertices. We describe the natural onetoone correspondence between sets of squarefree monomials of degree d and duniform hypergraphs with the following notation. For a finite duniform hypergraph H on V={x _{1},…,x _{ n }} and edge set E, each edge e _{ i } of H encodes a squarefree monomial \(x^{e_{i}}:=\prod_{j\in e_{i}} x_{j}\) of degree d in the polynomial ring k[x _{1},…,x _{ n }] over a field k. Thus, the hypergraph H can be written as a set of monomials representing the edges: \(H = \{x^{e_{1}}, \dots, x^{e_{k}}\}\). The edge subring of the hypergraph H, denoted by k[H], is the monomial subring of the polynomial ring generated by the edges of H; in symbols,
Letting \(t_{e_{i}}\) be a variable representing the edge e _{ i }, define a ring homomorphism \(\phi_{H} : k[t_{e_{i}}] \to k[H]\) with \(\phi_{H}(t_{e_{i}})=x^{e_{i}}\). The kernel of this monomial map, denoted by I _{ H }, is the toric ideal of the edge subring of the hypergraph H. It encodes the algebraic relations among the edges of the hypergraph.
A first problem of interest is to describe the combinatorics of generators of the ideal for an arbitrary hypergraph. With this in mind, we define monomial walks on hypergraphs and the balancing condition in Sect. 2, which are the natural direct generalization of monomial walks on graphs. Theorem 2.8 characterizes the binomials in I _{ H } in terms of such monomial walks, extending the classical theorems of Villarreal [22, 23], and Ohsugi and Hibi [13]. In Sect. 3 we outline the relevance of our results: the correspondence between statistical models and hypergraphs is explored in Sect. 3.1, and we derive a connection to the wellstudied settheoretical problem of combinatorial discrepancy in Sect. 3.2. Given the apparent difficulty of the problem of describing all primitive monomial walks on an arbitrary hypergraph, we dedicate Sect. 4 to sparsely intersecting hypergraphs, which generalize the three types of primitive walks on graphs. In particular, Propositions 4.2, 4.5, 4.9 and Theorem 4.12 describe several supporting hypergraphs of primitive monomial walks. Their basic building blocks consist of matchings and sunflowers, wellknown in the hypergraph literature. In Sect. 5 we show that many of the results for graphs cannot be generalized, and the expected degree bounds do not hold. Finally, Sect. 6 outlines some of the many open problems that we hope will be addressed by the combinatorics community.
Algebraic properties of toric ideals of hypergraphs such as normality, Cohen–Macaulayness, and geometry of the corresponding polytopes are all open problems of interest, but are beyond the scope of this paper. Degree bounds of generators are studied in [8]. The problem of finding these generators algorithmically and understanding the complexity of such algorithms remains open and would make a significant contribution to applied algebraic statistics.
Monomial walks on hypergraphs
We begin with an example in the case when H is 2uniform.
Example 2.1
Consider the simplest case when H is a graph, say, the complete graph H=K _{5} on vertices x _{1},…,x _{5}. The edge subring k[K _{5}]⊆k[x _{1},…,x _{5}] is parametrized by
The toric ideal \(I_{K_{5}}\) is generated by the following 10 quadrics:
For example, the binomial f:=t _{13} t _{24}−t _{12} t _{34}∈I _{ H } is represented in the leftmost graph of Fig. 1. Since f is represented by an even cycle with alternating colors on the edges, we say that it arises from the cycle.
In general, when H is a graph, then the generators of the ideal I _{ H } are known in terms of these redblue colorings on the edges of the graph:
Theorem 2.2
The toric ideal of the edge subring of a graph G is generated by binomials arising from (primitive) even closed walks on G.
Such binomials have been characterized by the following result which, in fact, provides a Graver basis for the toric ideal.
Theorem 2.3
Primitive even closed walks on G are one of the following: (i) even cycles, (ii) two odd cycles sharing a vertex, or (iii) two odd cycles such that there are two walks connecting a vertex in one cycle with a vertex in the other.
An even closed walk is the exact structure in a graph required to give rise to a binomial: the sequence of edges in the walk induces a bipartition of the edges with multiplicity and all vertices in the walk have the same degree in each of the two parts. We propose the following generalization of an even closed walk on a hypergraph. Following [23], we call it a monomial walk.
Definition 2.4
A monomial walk on a duniform hypergraph is an even sequence of edges \(\mathcal{W} := \{e_{1}, e_{2}, \dots, e_{2k}\}\), where each edge e _{ i } in the sequence intersects the subsequence {e _{1},e _{2},…,e _{ i−1}}, and each vertex \(v\in\bigcup_{i=1}^{2k} e_{i}\) covered by \(\mathcal{W}\) satisfies the following balancing condition: \( \{ e_{i} \in\mathcal{W} : v\in e_{i} \text{ and } i \text{ is odd}\}= \{ e_{i} \in\mathcal{W}: v\in e_{i} \text{ and } i \text{ is even}\}\).
Each monomial walk gives rise to a binomial in the ideal I _{ H }. Namely, we say that a binomial \(f_{\mathcal{W}}\) arises from \(\mathcal{W}\) if
A monomial walk is said to be primitive if there does not exist a proper subwalk, that is, a collection \(\mathcal{W}'\) of odd and evennumbered edges, properly contained in \(\mathcal{W}\), such that \(\mathcal{W}'\) is also a monomial walk.
For the remainder of the paper, we occasionally refer to a monomial walk as a walk. For convenience, all hypergraphs in this paper will be uniform. In the nonuniform case, binomials in I _{ H } are not necessarily supported by alternating monomial walks; although one can extend the balancing condition to the nonuniform case. Algebraically, we are simply considering homogeneous toric ideals.
Remark 2.5
(Bicoloring the edges in \(f_{\mathcal{W}}\))
The classical theorems for graphs use odd and even numbered edges to describe \(f_{\mathcal{W}}\). For hypergraphs, however, we have found this notation and related pictorial representations extremely tedious. Thus, from now on, we refer to the even and oddnumbered edges as blue and red edges. This bicoloring is implicit in the graphs case as well.
Example 2.6
Consider a complete 3uniform hypergraph on 12 vertices. Figure 2 shows three monomial walks on this hypergraph.
In general, there is no squarefree assumption on the binomial \(f_{\mathcal{W}}\). Walking along the same edge multiple times during \(\mathcal{W}\) is allowed, so it is possible that e _{ i }=e _{ j } for i≠j. (See Fig. 2(C) for an example.) If an edge in a monomial walk appears as both red and blue, then these two copies of the edge can be safely removed. The remaining set of edges constitutes another monomial walk. On the binomial side, this corresponds to a multiple of a smaller binomial.
Definition 2.7
Let \(\mathcal{E}\) be a (multiset) collection of an even number of edges in H. We say that \(\mathcal{E}\) is balanced with respect to a given bicoloring of H if for each vertex v covered by \(\mathcal{E}\), the number of red edges containing v equals the number of blue edges containing v; in symbols:
The constructions above imply the following.
Theorem 2.8
Let H be a uniform hypergraph. Any balanced collection of edges \(\mathcal{E}\subset E(H)\) constitutes a monomial walk, or a collection of disjoint monomial walks.
In particular, the ideal I _{ H } is generated by binomials \(f_{\mathcal{W}}\) arising from primitive monomial walks \(\mathcal{W}\) on H.
Proof
A binomial f is in the ideal I _{ H } if and only if the set of edges \(\mathcal{E}\) appearing in the support of the binomial is balanced. Namely, f∈I _{ H } is given by a collection of edges such that
where the coloring is induced by the binomial f itself. Being in the kernel of φ _{ H } is equivalent to
This, in turn, is equivalent to condition (⋆).
We need to show that any balanced collection of edges with respect to some bicoloring can be written as arising from a monomial walk, or a set of disjoint monomial walks. Throughout the proof we will maintain a sequence of visited edges \(\mathcal{W}\) from \(\mathcal{E}\), and the set of red and blue degrees of each vertex v covered by \(\mathcal{E}\), which we denote by \(\deg_{\mathrm{red}}(v;\mathcal{E})\) and \(\deg_{\mathrm{blue}}(v;\mathcal{E})\), respectively.
We begin with \(\mathcal{W}=\emptyset\). Pick an arbitrary blue edge \(e\in\mathcal{E}\), define e _{1}:=e, and update \(\mathcal{W}=\mathcal{W}\cup \{e_{1}\}\), \(\mathcal{E}=\mathcal{E}  \{e_{1}\}\), noting that \(\deg_{\mathrm{blue}}(v;\mathcal{E})\) has decreased by 1, for every v∈e _{1}. Pick an arbitrary u _{1}∈e _{1}. By (⋆), there exists a red edge \(e'\in\mathcal{E}\) such that u∈e′. Let e _{2}:=e′ and update \(\mathcal{W}=\mathcal{W}\cup\{e_{2}\}\), \(\mathcal{E}=\mathcal{E}  \{e_{2}\}\), noting that \(\deg_{\mathrm{red}}(v;\mathcal{E})\) has decreased by 1, for every v∈e _{2}. Clearly, u _{1} is now balanced in \(\mathcal{W}\).
Repeat recursively as follows. Check whether \(\deg_{\mathrm{blue}}(v;\mathcal{E})=\deg_{\mathrm{red}}(v;\mathcal{E})\) for all v. If not, there must exist an unbalanced vertex covered by \(\mathcal{W}\). For odd (resp. even) i, there exists a blue (resp. red) edge in \(\mathcal{E}\) that intersects an even (resp. odd) edge e _{ j } in \(\mathcal{W}\), j<i. Call this edge e _{ i } and move it to \(\mathcal{W}\).
On the other hand, if \(\deg_{\mathrm{blue}}(v;\mathcal{E})=\deg_{\mathrm{red}}(v;\mathcal{E})\) holds for every v, we have produced a monomial walk \(\mathcal{W}\). In the case where \(\mathcal{E}\) is not empty, the algorithm restarts on the current \(\mathcal{E}\) with empty \(\mathcal{W}\) to find additional edgedisjoint monomial walks. Finally, the algorithm stops when \(\mathcal{E}\) is empty.
What we have shown is that every binomial in the ideal I _{ H } arises from a finite collection of monomial walks. That only primitive binomials are necessary to generate the ideal follows from standard arguments: e.g. see Lemma 4.6 in [19]. □
Naturally, if a certain edge appeared multiple times in \(\mathcal{E}\), each occurrence of the edge gets a unique label, so that the above argument is valid for multisets of edges.
Recall that the Graver basis is a special generating set of the toric ideal, usually very large and algebraically redundant, but combinatorially nice. What we have shown above is that the Graver basis of I _{ H } consists of primitive monomial walks with support on the edge set of H. In addition, as in the graphs case, since the Graver basis contains all reduced Gröbner bases, we can say the following:
Corollary 2.9
If f is a polynomial in any reduced Gröbner basis of I _{ H }, then \(f=f_{\mathcal{W}}\) for some primitive monomial walk \(\mathcal{W}\).
The natural problem now is to understand primitive monomial walks on a given hypergraph H. We will concern ourselves with a criterion which will allow us to determine if a given set of edges is a primitive monomial walk. To this end, we make the following definition.
Definition 2.10
A multihypergraph \(\mathcal{H}\) is called a monomial hypergraph if its edges can be colored so that they are balanced (see Definition 2.7). Such an \(\mathcal{H}\) is primitive if there exists a balanced bicoloring that makes the corresponding monomial walk primitive.
The purpose of this definition is twofold. First, note that a monomial hypergraph corresponds to a monomial walk. However, a given walk comes equipped with the blue and red bicoloring of the edges (i.e., the binomial is given explicitly); whereas in a monomial hypergraph, the coloring is not fixed (i.e., it represents the support of the binomial). In the graph case, this distinction is less important because the coloring is more obvious. For example, when we say that “even cycles are generators of the toric ideal of a graph”, we do not specify how to color the edges of the cycle; however, upon the inspection of the proof, of course, it is obvious that the edges are colored in alternating colors to give rise to a binomial. In hypergraphs, such a choice of coloring may not always be immediate or unique, so this distinction is important. Second, a monomial hypergraph is not a simple hypergraph: there is no squarefree assumption on the monomial walks, so an edge in a hypergraph may be traced multiple times in a walk over its edge set. This generality is required since we cannot assume that the binomials in I _{ H } are squarefree.
Corollary 2.11
The set of binomials
is a universal Gröbner basis for the toric ideal I _{ H }.
In summary: a monomial walk has an alternating coloring on the edges and requires that all the visited vertices are balanced with respect to that coloring; and a monomial hypergraph is a multihypergraph with the supporting edge set being that of H, it corresponds to a monomial walk, but its coloring is not fixed. The monomial hypergraph corresponding to the monomial walk \(\mathcal{W}\) is the combinatorial signature of the binomial \(f_{\mathcal{W}}\in I_{H}\).
Monomial hypergraphs: from loglinear models to discrepancy
Before exploring the structure of monomial hypergraphs in the next section, let us briefly reflect on their relevance. We explain how they arise naturally in algebraic statistics, and proceed to show how they relate to combinatorial discrepancy of hypergraphs.
Parameter hypergraphs
Monomial maps make a notable appearance in statistics: they encode important families of statistical models for discrete random variables called loglinear models. A loglinear model is a family of joint probability distributions that are described by monomials in the model parameters (equivalently, minimal sufficient statistics of the model are in the row span of a linear map, and hence the name; but let us not dwell on this here). To see how monomial maps arise, let us consider the following very simple example. Two random variables Z _{1} and Z _{2}, taking a and b values, respectively, are said to be independent if the joint probability P _{ ij }:=Prob(Z _{1}=i,Z _{2}=j) factors as follows: P _{ ij }=Prob(Z _{1}=i)Prob(Z _{2}=j) for all i∈[a] and j∈[b]. Denote the probabilities Prob(Z _{1}=i) and Prob(Z _{2}=j) by x _{ i } and x _{ a+j }, respectively. The unknown quantities x _{1},…,x _{ a+b } are called the model parameters; they completely determine this family of distributions. We can summarize this simple example by saying that the independence model for Z _{1} and Z _{2} is specified by the monomial map that sends P _{ ij } to x _{ i } x _{ a+j }. In this sense, the edge subring k[K _{ a,b }] of the complete bipartite graph K _{ a,b } on the vertex set {x _{1},…,x _{ a }}⊔{x _{ a+1},…,x _{ a+b }} describes the structure of the joint probability distributions that belong to the independence model.
This correspondence can be naturally extended to any more complicated model for discrete random variables Z _{1},…,Z _{ m }, whose joint probabilities are parametrized by squarefree monomials in the model parameters x _{1},…,x _{ n }. Each such loglinear model \(\mathcal{M}\) is encoded by a hypergraph \(H_{\mathcal{M}}\) on the vertex set x _{1},…,x _{ n }, constructed as follows: {x _{ j }}_{ j∈J } is an edge of \(H_{\mathcal{M}}\) if and only if the index set J describes one of the joint probabilities in the model; that is, there exist values i _{1},…,i _{ m } such that Prob(Z _{1}=i _{1},…,Z _{ m }=i _{ m })=∏_{ j∈J } x _{ j }.
Henceforth, we will name \(H_{\mathcal{M}}\) the parameter hypergraph of the model \(\mathcal{M}\). Somewhat surprisingly, toric ideals \(I_{H_{\mathcal{M}}}\) are quite relevant for testing goodness of fit of the model \(\mathcal{M}\). Without going into statistical details, a nonasymptotic test for model/data fit is carried out by exploring the fiber of a given point in the image of the monomial map \(\phi_{H_{\mathcal{M}}}\). In terms of the hypergraph \(H_{\mathcal{M}}\), an observed data point corresponds to a collection of edges; for convenience, think of them as colored blue. The fiber of that data point then consists of all red edge sets that balance it, and thus have the same image under the map \(\phi_{H_{\mathcal{M}}}\). The recipe for exploring a fiber is to move from any point to another point by applying a Markov move, which corresponds to a generator of the toric ideal \(I_{H_{\mathcal{M}}}\) of the model. Since the binomials in \(I_{H_{\mathcal{M}}}\) are supported by balanced edge sets, this move is interpreted as removing the observed (blue) edges and replacing them by the red edges, arriving at another monomial with the same image. The correspondence between Markov moves and toric generators is wellknown in algebraic statistics literature, and often referred to as the fundamental theorem of Markov bases. The technical description of how such a random walk on a fiber is actually implemented is beyond the scope of this paper; but the interested reader is referred to [5] and the recent text [1].
In general, finding Markov bases for many relevant statistical modes is a wideopen problem; some complexity bounds and a few structural theorems are known. Therefore, our goal is to understand the combinatorial structure of monomial hypergraphs supported on \(H_{\mathcal{M}}\). Instead of focusing on one loglinear model, we explore combinatorial signatures of general monomial hypergraphs. We focus on homogeneous ideals, thus the models parametrized by uniform hypergraphs. Finally, as explained in [1], for homogeneous toric models with statistical sampling constraints or with structural zeros in the model, a minimal generating set of the toric ideal may not suffice for connecting the fibers, and instead squarefree part of the Graver basis is used.
As mentioned in the introduction, the edge subring of a graph has appeared in several results in algebraic statistics. By defining the parameter hypergraph \(H_{\mathcal{M}}\), which the reader should agree is quite a natural generalization, we give an applied motivation to extend the wellknown results on toric ideals of graphs to the more general case. By Theorem 2.8, Markov moves for any (homogeneous) loglinear model \(\mathcal{M}\) are encoded by primitive monomial hypergraphs supported on a (uniform) \(H_{\mathcal{M}}\). One of our main goals is to frame this problem in terms of combinatorics of hypergraphs.
A dual problem: combinatorial discrepancy
To that end, we begin by noting a relation between monomial hypergraphs and zerodiscrepancy hypergraphs. Broadly, the combinatorial discrepancy problem asks for a coloring of the vertices of a hypergraph H=(V,E) such that each hyperedge contains roughly the same number of vertices of each color. (The reader is directed to [2, 11] for a detailed introduction to discrepancy theory.) Let us formally define the discrepancy of a hypergraph.
Definition 3.1
The discrepancy of a hypergraph H=(V,E), denoted \(\operatorname {disc} (H)\), is the quantity \(\operatorname {disc} (H)=\min_{\chi: V\to \pm1} \max_{e\in E} \sum_{v\in e} \chi(v)\). In words, it is the minimum, over all colorings χ, of the maximum discrepancy of an edge in H.
Given a hypergraph H, the dual hypergraph H ^{∗} is the hypergraph containing a vertex v ^{∗} for each edge e∈E(H) and an edge e ^{∗} for each v∈V(H); here e ^{∗}={v ^{∗} ∈ V(H ^{∗}):v ^{∗} corresponds to an edge e ∈ E(H) that is incident in H to vertex v}. More briefly, the vertexedge incidence matrix of H ^{∗} is the transpose of the vertexedge incident matrix of H.
It is not hard to see that a zerodiscrepancy hypergraph H ^{∗}=(V ^{∗},E ^{∗}) is dual to a monomial hypergraph H=(V,E): the mapping χ:V ^{∗}→±1 acts on the vertices of H ^{∗} but it acts on the edges of H. Zero discrepancy for H ^{∗} implies that each edge of H ^{∗} contains the same number of red and blue vertices and, as a consequence, each vertex of H is incident to the same number of red and blue edges. This establishes the following theorem.
Theorem 3.2
A hypergraph H is a monomial hypergraph if and only if \(\operatorname {disc} (H^{*})=0\). Furthermore, a monomial hypergraph H is primitive if and only if no coloring χ:V ^{∗}→±1 realizing \(\operatorname {disc} (H^{*})=0\), also induces a zero discrepancy coloring on a (nonempty) subhypergraph \(\tilde{H}\) of H ^{∗} produced by deleting any subset of vertices and shrinking the edges.
Deleting a vertex from H ^{∗} and shrinking its incident edges is equivalent to deleting an edge in the dual hypergraph H. There are some other interesting facts here. While a monomial hypergraph can have multiple edges, this is not true or necessary for the dual zerodiscrepancy hypergraph, as multiple edges in \(\mathcal{H}\) would correspond to vertices that are incident to the same set of edges in the dual.
An important inapproximability result of [4] on discrepancy implies that it is NPHard to decide whether a hypergraph H=(V,E) is the support for a monomial hypergraph, or whether for every bicoloring of E(H), there exists a vertex v∈V(H) such that \(\vert \deg_{\mathrm{red}}(v)\deg_{\mathrm{blue}}(v)\vert \geq \varOmega(\sqrt{E(H))}\).
Given the difficulty of deciding the discrepancy of a hypergraph, we pursue a structural theorem for monomial hypergraphs in the next section.
Sparse bouquets
Theorem 2.8 can be used to give a characterization of (primitive) monomial walks in terms of the existence of an edge partition satisfying the degree condition (⋆). However, it gives no other information about the structure of the supporting hypergraphs.
In this section, we use Theorem 2.8 to construct and study supports of monomial hypergraphs that parallel the graphs case. For example, consider case (ii) from Theorem 2.3. Two odd cycles glued at a vertex support a primitive monomial walk on a graph. Increasing the degree of that core vertex could produce monomial walks, but they can never be primitive. In contrast, the core vertices in hypergraph supports of primitive walks can have degree larger than 2; in fact, it can be arbitrarily high!
The first three results in this section directly generalize cases (i), (ii) and (iii) (for short cycles) from Theorem 2.3. First, a pair of perfect matchings generalizes an even cycle, and its supporting hypergraph is characterized in Proposition 4.2. Second, a monomial sunflower generalizes a bowtie (two 3cycles sharing a vertex), and its supporting hypergraphs are characterized in Proposition 4.5. Finally, a partitioned core sunflower generalizes two 3cycles connected by two paths, and the supporting hypergraphs are described in Proposition 4.9.
A further generalization summarizing the above propositions is given in Theorem 4.12. So, unlike in the case of graphs, these three results do not exhaustively characterize monomial hypergraphs. Nevertheless, starting from sunflowers and matchings, natural building blocks of monomial hypergraphs, we can obtain much more complex structures. We delve more into the open problem of structurally characterizing primitive monomial hypergraphs in Sect. 6.
Matchings
Definition 4.1
A matching on a hypergraph H=(V,E) is a subset M⊂E of independent edges, that is, no two edges intersect. A matching is called perfect if it covers all the vertices of the hypergraph, i.e., V(M)=V.
The simplest monomial hypergraph can be formed using a pair of perfect matchings on the same set of variables. Next we give a sufficient and necessary condition for the primitivity of such a monomial hypergraph. A hypergraph H=(V,E) is said to be connected if its primal graph is connected, where the primal graph has the same vertex set and an edge between any two vertices contained in the same hyperedge.
Proposition 4.2
(Pair of matchings)
Let H=(V,E) be a duniform hypergraph such that E=M _{ r }⊔M _{ b }, where M _{ r }, M _{ b } form two edgedisjoint perfect matchings on the vertex set V. A monomial hypergraph \(\mathcal{H}\) with support H is primitive if and only if H is connected and \(\mathcal{H}\) contains no multiple edges.
Proof
Let \(\mathcal{H}\) be a primitive monomial hypergraph and consider a primitive monomial walk \(\mathcal{W}\) on the edges of \(\mathcal{H}\) with an appropriate associated bicoloring. Any walk is connected, so suppose for contradiction that edge e∈E has multiplicity k in \(\mathcal{W}\). By construction of the supporting hypergraph H, there exist exactly two distinct edges in H, say e and f, containing any vertex v. We may assume that all copies of e in \(\mathcal{W}\) are red, as two copies of e in \(\mathcal{W}\) that belong to a different color partition would create a trivial subwalk. By the degree condition (⋆) there must be k copies of f in \(\mathcal{W}\), all colored blue. Edges e and f are distinct so there exist a vertex u in f and a unique edge g in H such that g contains u and has multiplicity k in \(\mathcal{W}\). But H is connected, so we may repeat this process for each vertex in \(\mathcal{W}\) to conclude that all edges in E appear in \(\mathcal{W}\) in k copies of the same color, contradicting the primitivity of \(\mathcal{W}\).
For sufficiency, we need only produce a bicoloring for the edges of H, taken with no multiplicity, such that the associated walk is primitive. We construct the required bicoloring by coloring edges in M _{ r } red and edges in M _{ b } blue. Assume by way of contradiction that there exists a smaller walk \(\mathcal{W}\) in \(\mathcal{H}\) that is also a monomial walk. Let \(V_{\mathcal{W}}\) and \(E_{\mathcal{W}}\) be the vertex and edge sets induced by the walk \(\mathcal{W}\), and let \(\mathcal{H}_{\mathcal{W}}=(V_{\mathcal{W}},E_{\mathcal{W}})\). It is clear that every vertex in any monomial walk, and so specifically in \(\mathcal{H}_{\mathcal{W}}\) must have degree at least 2 (the sum of the blue degree and the red degree). Since \(\mathcal{W}\) is a subwalk, there must be some edge or vertex of \(\mathcal{H}\) that \(\mathcal{W}\) does not contain. But every vertex in \(H_{\mathcal{W}}\) has degree 2 so there can be no more edges in \(\mathcal{H}\setminus \mathcal{H}_{\mathcal{W}}\) containing any vertex from \(\mathcal{H}_{\mathcal{W}}\). Then H is disconnected. □
Figures 2(A) and 2(B) are primitive matchings. The former is also commonly called a tight cycle.
Monomial sunflower
As pairs of matchings have the property that each vertex has degree 2, they generalize even cycles on graphs. In contrast, the next case in Theorem 2.3 has a distinguished vertex of higher degree. Its natural generalization in hypergraphs, which we will call a monomial sunflower, is the topic of this section, and is based on a very useful, highly structured hypergraph, the sunflower, which, incidentally, is guaranteed to occur in hypergraphs with large enough edge sets, independently of the size of the vertex set. (See e.g. [10].)
Definition 4.3
A duniform hypergraph H=(V,E) is a sunflower if e _{ i }∩e _{ j }=C for all edges e _{ i }≠e _{ j }∈E, and C⊂V. The set of vertices C is called the core of the sunflower, and each e _{ i } is called a petal.
In general, one may allow the core to be empty, in which case you simply get a matching. In this paper, we will explicitly use the name matching instead of allowing for empty cores.
For the remainder of the paper, fix the following notation.
Notation 4.4
For a subset C⊂V(H), define H−C to be the hypergraph with vertex set (V∖C) and edges {e _{ i }∖C:e _{ i }⊂E(H)}. Similarly, for a subset \(C\subset V(\mathcal{H})\), define \(\mathcal{H}C\) to be the multihypergraph with vertex set (V∖C) and edges \(\{e_{i} \setminus C: e_{i}\subset E(\mathcal{H})\}\). The hypergraphs H−C and \(\mathcal{H}C\) are not necessarily uniform and may include empty edges, or multiple copies of an edge. See Figs. 3, 4 and 9 for examples.
Clearly, there is no nontrivial monomial walk on a sunflower: since the degree of any noncore vertex is 1, every petal needs to be both blue and red, resulting in the zero binomial. Thus, a sunflower alone cannot be a support of a monomial hypergraph. The noncore petal vertices could, however, be balanced by a perfect matching. To that end, consider H to be a sunflower with a perfect matching on its noncore vertices. In Proposition 4.5, we show that a primitive monomial hypergraph with support H can be characterized by counting petals.
Consider the set of connected components
For convenience, define \(G_{\mathcal{J}}:=\bigcup_{j\in\mathcal{J}}G_{j}\), for any index set \(\mathcal{J}\subseteq \mathcal{I}\). By abuse of notation we will write G _{ j } for the number of edges in the multiset {e∈H:e∩C≠∅ and e∖C∈G _{ j }}.
The next proposition is based on the idea that counting petals detects balanced walks on H.
Proposition 4.5
(Sunflower)
Let H be a hypergraph consisting of a sunflower and a perfect matching on the noncore vertices of the sunflower. The hypergraph H is the support of a monomial hypergraph \(\mathcal{H}\) if and only if there exists a partition \((G_{\mathcal{J}}, G_{\mathcal{K}})\) of the connected components of H−C such that
In addition, \(\mathcal{H}\) is primitive if and only if the partition \((G_{\mathcal{J}}, G_{\mathcal{K}})\) also satisfies the following two conditions:

(i)
no two subsets \(\mathcal{J}'\subsetneq \mathcal{J}\) and \(\mathcal{K}'\subsetneq \mathcal{K}\) satisfy \(\sum_{j\in \mathcal{J}'} m'_{j}G_{j}=\sum_{k\in \mathcal{K}'} m'_{k}G_{k}\) for some integers \(m'_{j}\leq m_{j}\), \(m'_{k}\leq m_{k}\) and

(ii)
there are no integers \(m'_{j}, m'_{k}\in\mathbb{Z}\) such that the partition \((G_{\mathcal{J}}, G_{\mathcal{K}})\) of the connected components of H−C satisfies \(\sum_{j\in \mathcal{J}} m'_{j}G_{j}=\sum_{k\in \mathcal{K}} m'_{k}G_{k}\), where \(m'_{j}\leq m_{j}\), \(m'_{k}\leq m_{k}\) and for at least one integer the inequality is strict.
Proof
For the easy direction of the first statement, suppose \((G_{\mathcal{J}}, G_{\mathcal{K}})\) is a partition of the connected components of H−C such that \(\sum_{j\in \mathcal{J}} m_{j}G_{j}=\sum_{k\in \mathcal{K}} m_{k}G_{k}\) for some \(m_{j},m_{k}\in\mathbb{Z}\). Construct the monomial walk \(\mathcal{W}\) as follows: for every edge e having a nonempty intersection with \(G_{j}, j\in \mathcal{J}\) (respectively, \(G_{k}, k\in\mathcal{K}\)), include in \(\mathcal{W}\), m _{ j } (respectively, m _{ k }) copies of edge e. Color the matching edges red for \(\mathcal{J}\) and blue for \(\mathcal{K}\), and use the opposite colors for the petal edges. This immediately ensures that the noncore vertices satisfy the degree condition (⋆). The degrees of a core vertex v are \(\deg_{\mathrm{blue}}(v)=\sum_{j\in \mathcal{J}} m_{j}G_{j}\) and \(\deg_{\mathrm{red}}=\sum_{k\in \mathcal{K}} m_{k}G_{k}\), respectively.
For the other direction of the first statement, suppose \(\mathcal{H}\) is a monomial hypergraph over H and consider the monomial walk \(\mathcal{W}\) consisting of the edges of \(\mathcal{H}\). For every vertex v in a component G _{ i } of H, there are exactly two distinct edges containing v: a petal edge e and a matching edge f. Suppose e appears with multiplicity m _{ i } in \(\mathcal{W}\). Without loss of generality, we may assume that all m _{ i } copies of e must belong to the same color partition, say red. By the degree condition (⋆), edge f must also appear in \(\mathcal{W}\) with multiplicity m _{ i }, and all m _{ i } copies must belong to the blue color partition. Now there exists a vertex u∈f∩e′ (u≠v) where e′ (e′≠e) is the unique petal edge containing u. To ensure u is balanced, by the degree condition, e′ must also appear in \(\mathcal{W}\) with multiplicity m _{ i }. Repeating this argument over all vertices in the connected component G _{ i } we conclude that each matching edge in G _{ i } and each petal edge whose noncore vertices fall in G _{ i }, must appear in \(\mathcal{W}\) with multiplicity m _{ i }. Then we may construct the partition \((G_{\mathcal{J}}, G_{\mathcal{K}})\) so that \(G_{i}\in G_{\mathcal{J}}\) if the matching edges it contains are colored red, and \(G_{i}\in G_{\mathcal{K}}\) otherwise. Since the core vertices are also balanced, the equation \(\sum_{j\in \mathcal{J}} m_{j}G_{j}=\sum_{k\in \mathcal{K}} m_{k}G_{k}\) holds, where the sum on the left hand side counts the blue degree of a core vertex and the right hand side its red degree.
The second statement in the proposition (primitivity) follows from the first statement and the definition of primitivity. Condition (i) describes all subwalks not using all edges of the supporting hypergraph H and condition (ii) describes all subwalks using all edges of H with smaller multiplicity than in \(\mathcal{H}\). □
We may now make the following definition.
Definition 4.6
A monomial sunflower is a monomial hypergraph with support set consisting of a sunflower and a perfect matching on the noncore vertices.
The partition \((G_{\mathcal{J}},G_{\mathcal{K}})\) corresponds to a bicoloring of the petals of the sunflower, as illustrated in the following example.
Example 4.7
Figure 3 demonstrates that H _{1} is the support of a simple nonprimitive monomial sunflower and its components. Each edge appears exactly once in \(\mathcal{H}_{1}\), and it is easy to see that for any two components in H−C, there is a bicoloring of \(\mathcal{H}_{1}\) such that the two components form a subwalk.
Figure 4 demonstrates a bicoloring of the monomial sunflower \(\mathcal{H}_{2}\) that makes it primitive. The appropriate multiplicities for each component are m _{1}=1 and m _{2}=2.
Independent petal bouquets
To produce more general monomial hypergraphs and characterize primitivity on them, we may generalize the monomial sunflower by considering a hypergraph H formed by a collection of sunflowers and a perfect matching on the subset of vertices that are not in the core of any sunflower. Let us, first, assume that the sunflowers are entirely disjoint.
Let H be a hypergraph consisting of a collection of vertexdisjoint sunflowers S _{1},S _{2},…,S _{ ℓ } and a perfect matching on the noncore vertices of ⋃_{ i } S _{ i }. We will call such an H a matchedpetal partitionedcore sunflower.
As we saw in Proposition 4.5, when H contains a single sunflower, the components of H−C and their multiplicities determine the degrees of the vertices of H and ensure that the core is balanced. In a matchedpetal partitionedcore sunflower, not all the components contribute to every sunflower S _{ i }. So, one needs to keep track of the components of H−C that contain vertices of S _{ i }, for each S _{ i }. We employ the following notation.
Notation 4.8
Let H be a matchedpetal partitionedcore sunflower. Given a fixed partition \((G_{\mathcal{J}}, G_{\mathcal{K}})\) of the connected components of H−C, and a fixed sunflower S _{ i } of H, define \(\mathcal{J}(S_{i})\subset \mathcal{J}\) (respectively, \(\mathcal{K}(S_{i})\subset \mathcal{K}\)) to include every \(j\in \mathcal{J}\) (respectively, \(j\in \mathcal{K}\)), such that G _{ j } contains a vertex of S _{ i }. In symbols, \(\mathcal{J}(S_{i})\) is the set of indices defined as \(\mathcal{J}(S_{i}):=\{j\in\mathcal{J}(S_{i}): j\in \mathcal{J} \text{ and } e\cap G_{j}\neq \emptyset \text{ for some edge } e\in S_{i}\}\).
Proposition 4.9
(Partitioned core sunflower)
Let H be a matchedpetal partitionedcore sunflower, and adopt Notation 4.8.
The hypergraph H is the support of a monomial hypergraph \(\mathcal{H}\) if and only if there exists a partition \((G_{\mathcal{J}}, G_{\mathcal{K}})\) of the connected components of H−C such that for each 1≤i≤ℓ:
In addition, \(\mathcal{H}\) is primitive if and only if the partition \((G_{\mathcal{J}}, G_{\mathcal{K}})\) also satisfies the following two conditions:

(i)
no two subsets \(\mathcal{J}'\subsetneq \mathcal{J}\) and \(\mathcal{K}'\subsetneq \mathcal{K}\) satisfy for each 1≤i≤ℓ,
$$\sum_{j\in \mathcal{J}(S_i)\cap \mathcal{J}'} m'_jG_{j}= \sum_{k\in \mathcal{K}(S_i)\cap \mathcal{K}'} m'_kG_{k}, $$for some integers \(m'_{j}\leq m_{j}\), \(m'_{k}\leq m_{k}\), and

(ii)
there are no integers \(m'_{j}, m'_{k}\in\mathbb{Z}\) such that the partition \((G_{\mathcal{J}}, G_{\mathcal{K}})\) of the connected components of H−C satisfies for each 1≤i≤ℓ,
$$\sum_{j\in \mathcal{J}(S_i)} m'_jG_{j}= \sum_{k\in \mathcal{K}(S_i)} m'_kG_{k}, $$where \(m'_{j}\leq m_{j}\), \(m'_{k}\leq m_{k}\) and for at least one integer the inequality is strict.
Proof
The first statement follows with the same argument as Proposition 4.5 for the noncore vertices. For a core vertex, given a partition \((G_{\mathcal{J}}, G_{\mathcal{K}})\) of the connected components of H−C, the sum \(\sum_{j\in \mathcal{J}(S_{i})} m_{j}G_{j}\) (respectively, \(\sum_{k\in \mathcal{K}(S_{i})} m_{k}G_{k}\)) counts the number of petals of sunflower S _{ i } whose remnants fall in components \(G_{\mathcal{J}(S_{i})}\) (respectively, \(G_{\mathcal{K}(S_{i})}\)), and as such completely determines the quantity that each part of the bipartition contributes to the degree of a core vertex v∈S _{ i }. As before, the second statement describes two types of subwalks. □
Example 4.10
Figure 5 demonstrates a primitive partitioned core monomial sunflower \(\mathcal{H}\). Note that all of the multiplicities are equal to one.
We relax the disjointness property of the sunflowers above, and allow the sunflowers to intersect on core vertices. Let H be a hypergraph consisting of a collection of sunflowers S _{1},S _{2},…,S _{ ℓ }, which may only intersect at their cores (known as a relaxed core sunflower), and a perfect matching on the noncore vertices of the sunflowers. We will call such an H a matchedpetal relaxedcore sunflower.
In the partitioned core case, the degree of each core vertex depends on exactly one sunflower. In contrast, allowing intersections in the cores causes the degree of a core vertex to depend on several sunflowers. This prompts the following notation.
Notation 4.11
For a matchedpetal relaxedcore sunflower, define \(\mathcal{I}(v)\) to consist of the set of indices i such that vertex v is in the core of sunflower S _{ i }. In symbols \(\mathcal{I}(v) := \{i\in\mathcal{I}(v):v\in S_{i}\}\).
In addition, Notation 4.8 extends to H.
Theorem 4.12
(Relaxed core sunflower)
Let H be a matchedpetal relaxedcore sunflower, and adopt Notations 4.8 and 4.11.
The hypergraph H is the support of a monomial hypergraph \(\mathcal{H}\) if and only if there exists a partition \((G_{\mathcal{J}}, G_{\mathcal{K}})\) of the connected components of H−C such that for each vertex v in the core of a sunflower:
In addition, \(\mathcal{H}\) is primitive if and only if the partition \((G_{\mathcal{J}}, G_{\mathcal{K}})\) also satisfies the following two conditions:

(i)
no two subsets \(\mathcal{J}'\subsetneq \mathcal{J}\) and \(\mathcal{K}'\subsetneq \mathcal{K}\) satisfy for each vertex v in the core of the sunflower,
$$\sum_{i\in \mathcal{I}(v)}\sum_{j\in \mathcal{J}(S_i)\cap \mathcal{J}'} m'_jG_{j}=\sum _{s\in \mathcal{I}(v)}\sum_{k\in \mathcal{K}(S_i)\cap \mathcal{K}'} m'_kG_{k}, $$for some integers \(m'_{j}\leq m_{j}\), \(m'_{k}\leq m_{k}\), and

(ii)
there are no integers \(m'_{j}\leq m_{j}\), \(m'_{k}\leq m_{k}\in\mathbb{Z}\), where the inequality is strict for at least one integer, such that the partition \((G_{\mathcal{J}}, G_{\mathcal{K}})\) of the connected components of H−C satisfies for each vertex v in the core of a sunflower,
$$\sum_{i\in \mathcal{I}(v)}\sum_{j\in \mathcal{J}(S_i)} m'_jG_{j}=\sum _{i\in \mathcal{I}(v)}\sum_{k\in \mathcal{K}(S_i)} m'_kG_{k}. $$
Proof
Follow the proof of Proposition 4.9 with one important exception. The degree of a core vertex may now depend on several sunflowers, and so the quantity necessary to calculate the degree of a core vertex v in part \(\mathcal{J}\), for example, is \(\sum_{i\in \mathcal{I}(v)}\sum_{j\in \mathcal{J}(S_{i})} m'_{j}G_{j}\). □
Proposition 4.9 can be obtained as a corollary of Theorem 4.12.
Example 4.13
Figure 6 demonstrates a simple primitive relaxed core monomial sunflower \(\mathcal{H}_{4}\) with four sunflowers. Each multiplicity equals one.
Bouquet complexity
The types of Graver elements studied in Propositions 4.2, 4.5 and 4.9 are, in fact, the simplest ones that may occur beyond the graphs case. They are the natural direct generalization of the three types of monomial walks on graphs from Theorem 2.3 (see also Fig. 1), each of which is an example of a sparse bouquet. By allowing the edges to contain more than two vertices, we inadvertently and significantly increase the complexity of the primitive walks. This already occurs on 3uniform hypergraphs and on sparse bouquets. We summarize some of the different ways in which the complexity has increased through the following examples.
In the case of graphs, it is known (see [23, Proposition 8.1.6]) that the largest degree of a vertex in the support of a primitive walk is 4, and that the edges in the support of the walk can be traversed at most twice. None of these restrictions extend to duniform hypergraphs (d>2), where a primitive walk can contain vertices with arbitrarily large degree; can contain arbitrarily many vertices of large degree; and a supporting edge can be used arbitrarily many times in a primitive walk.
It is not hard to see that there exists exactly one primitive duniform monomial sunflower with core of size d−1: the one containing exactly 2d+2 hyperedges. On the other hand, Fig. 7 demonstrates a family of primitive 3uniform bouquets on n vertices with number of edges of the order of n and containing a vertex with degree of the order of n. Figure 8 demonstrates a family of 3uniform bouquets in which the number of cores, vertex degrees, and edge multiplicities can be arbitrarily high.
Conclusion and open problems
Using the sparse hypergraphs as a starting point, one should consider the problem of classifying supports of more general monomial hypergraphs. However, this problem is highly nontrivial. For example, Fig. 9 represents a simplest example of a nonprimitive monomial walk that is not of the form covered in Sect. 4. The hypergraph consists of two overlapped walks. It is tempting to mimic the constructions for sparse bouquets and express this hypergraph as a union of, say, four sunflowers S _{1},…,S _{4} with two petals each: S _{1}={e _{1},e _{3}}, S _{2}={e _{2},e _{4}}, S _{3}={e _{5},e _{7}}, S _{4}={e _{6},e _{8}}. Then, removing the cores from these sunflowers and counting the petal remnants in the connected components, with multiplicity 1, as suggested by a direct generalization of Theorem 4.12, would reveal that S _{3}∪S _{4} is a balanced set of edges. In this way, one could detect that the monomial hypergraph S _{1}∪…∪S _{4} is not primitive.
In general, however, an arbitrary monomial hypergraph consists of sunflowers that overlap in unrestricted ways, and thus it is not guaranteed that removing the cores will disconnect and detect a subwalk. In particular, a direct generalization of the results from Sect. 4 does not hold without further assumptions.
Problem 6.1
Find a criterion to detect primitivity of a given monomial hypergraph H based on a subset of the edges in H.
At the moment, the best result in this direction is Theorem 2.8. For “sparse” collections of sunflowers and matchings, Sect. 4 offers more efficient criteria. But if the hypergraph is denser, an arbitrary sunflower decomposition alone does not seem to suffice.
More generally, it would be interesting, although difficult, to solve the following specific instance of the general problem of characterizing primitive walks.
Problem 6.2
Characterize (combinatorially) the supports of primitive monomial hypergraphs over an arbitrary 3uniform hypergraph.
We remind the reader that even for 3uniform hypergraphs, any of the types of bouquets that we describe can appear in the Graver basis, as demonstrated throughout the paper. Thus Problem 6.2 is highly nontrivial; however, we expect it to have a nice answer for a subclass of 3uniform hypergraphs, for example, where it is possible to partition a vertex set in a way that restricts the types of bouquets that may appear. On the other hand, recall the ultimate goal of interest to statistical applications: we are looking for characterizations of the structure of monomial walks for a family of hypergraphs specified by conditions that are significantly more restrictive than uniformity. The family will come from a statistical model with a prescribed edge size, or edge types. Such conditions will necessarily restrict the ways in which edges of H can intersect, and thus may imply the sparsity of walks supported on H.
Furthermore, one can quickly derive the entire Graver basis for the following two classes of hypergraphs. For any 2regular hypergraph, it consists only of pairs of matchings as in Proposition 4.2. In particular, the Graver basis of a connected 2regular hypergraph contains exactly one element. Secondly, consider the duniform hypergraph \(C_{r}^{d}=(V,E)\), produced by letting V=⨆_{1≤i≤r } V _{ i }, with each V _{ i } having size d/2 (d even), and edge set E={V _{ i }∪V _{ j }: for 1≤i<j≤r}. A (primitive) monomial walk on \(C_{r}^{d}\) corresponds to a (primitive) even closed walk in the complete graph K _{ r }, as each hyperedge in \(C_{r}^{d}\) is in a onetoone correspondence with an edge in K _{ r }.
Our next problem concerns generating primitive monomial hypergraphs. It is possible to generate such hypergraphs by contraction or “gluing” operations using known monomial hypergraphs (and sparse bouquets in particular) as building blocks. For example, one may produce duniform monomial sunflowers from a duniform pair of perfect matchings by identifying selected sets of independent vertices. This operation preserves primitivity. The converse is only true if vertexidentification takes place within connected components. In general, one may also glue two hypergraphs H _{1}, H _{2} along the vertices of an edge to produce a new hypergraph H, introducing new primitive elements in I _{ H } constructed from binomials from \(I_{H_{1}}\) and \(I_{H_{2}}\). The support of the monomial hypergraph in Fig. 10 was partially constructed from the support of four monomial sunflowers with this gluing operation. It is not hard to see that \(\mathcal{B}\) is a monomial hypergraph.
A general gluing operation for graphs, based on a toric fiber product, is described in [7]. It preserves many properties of toric ideals.
Problem 6.3
Find a gluing operation on hypergraphs that preserves primitivity of monomial walks. Is it possible to generate all primitive monomial hypergraphs starting from sparse bouquets?
The primitive walks in hypergraphs are clearly much more general than in the case of graphs, as Sect. 5 shows. Nevertheless, we expect many of the other properties of the coordinate ring of I _{ H } to have combinatorial interpretations. For example, the first author studies the degree bounds on the minimal generators of I _{ H } in [8], generalizing the theorems known for graphs.
Finally, the problems of finding nice term orders and reduced Gröbner bases of I _{ H }, characterizing Cohen–Macaulayness and normality of the coordinate ring and the corresponding polytope, and relating known coloringinspired properties of hypergraphs to various invariants of the toric ideal I _{ H } are wide open.
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Acknowledgements
The authors thank György Turán for inspiring discussions, at the start of this project, about the fundamental problem of walks in hypergraphs; and also Elizabeth Gross for suggesting the name “bouquet”, and proofreading an early version of the manuscript. The second author is grateful to Amitava Bhattacharya for introducing her to combinatorial discrepancy. Finally, we are grateful to the anonymous referee for thoughtful and thorough comments that helped improve this work.
The authors acknowledge support by grant FA95501210392 from the U.S. Air Force Office of Scientific Research (AFOSR) and the Defense Advanced Research Projects Agency (DARPA). The AMS Simons Travel Grant has helped the first author tremendously during the first year of the project.
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Petrović, S., Stasi, D. Toric algebra of hypergraphs. J Algebr Comb 39, 187–208 (2014). https://doi.org/10.1007/s108010130444y
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Keywords
 Hypergraph
 Edge ring
 Markov basis
 Toric ideal
 Monomial walk
 Sunflower
 Combinatorial discrepancy
 Parameter hypergraph