Hypergraph Colorings, Commutative Algebra, and Gröbner Bases

A uniform hypergraph is properly k-colorable if each vertex is colored by one of k colors and no edge is completely colored by one color. In 2008 Hillar and Windfeldt gave a complete characterization of the k-colorability of graphs through algebraic methods. We generalize their work and give a complete algebraic characterization of the k-colorability of r−uniform hypergraphs. In addition to general k colorability, we provide an alternate characterization for 2-colorability and apply this to some constructions of hypergraphs that are minimally non-2colorable. We also provide examples and verification of minimality for non-2-colorable 5and 6-uniform hypergraphs. As a further application, we give a characterization for a uniform hypergraph to be conflict-free colorable. Finally, we provide an improvement on the construction introduced by Abbott and Hanson in 1969, and improved upon by Seymour in 1974.


Introduction
Many problems in combinatorics have elegant algebraic characterizations and many useful tools in combinatorics are based on algebraic methods. Such tools include the graph polynomial, the Combinatorial Nullstellensatz, and the Stanley-Reisner ideal. These tools allow alternative methods for analyzing combinatorial properties of graphs and hypergraphs by encoding them into polynomial ideals and algebraic varieties. The focus of this thesis is the use of tools from commutative algebra, namely Gröbner bases and radical ideals, to provide a complete algebraic characterization for general colorability of uniform hypergraphs. We also provide computationally supported bounds on specific types of hypergraph colorings. This thesis uses results from commutative algebra, algebraic geometry, enumerative combinatorics, and graph theory. We utilize ideas and tools including: polynomial ideals over algebraically closed fields, zero-dimensional algebraic varieties, Gröbner bases, integer partitions, and Hilbert's Nullstellensatz. Traditional Graph Theory is the study of 2-uniform hypergraphs. Although some techniques and applications developed in graph theory have been generalized to hypergraphs many of the techniques and theorems do not directly generalize.
Hypergraph Theory is an active area in combinatorics, both in direct research, and in application to other areas of mathematics. One of the most thoroughly studied area of graph theory is that of graph colorings. Thus it is natural to generalize the idea of a graph coloring to that of a hypergraph coloring. Definition 1.2. Let k be a positive integer and let H be a uniform hypergraph. A proper k−coloring of a hypergraph H is a map, c, from the vertex set of H to a set of k colors, C: where each vertex is assigned exactly one color, and no edge is colored completely by a single color.
There are many applications of hypergraph colorings in diverse areas such as Computer Science, Statistical Physics, and Mathematical Chemistry. It is known that deciding whether an r-uniform hypergraph is k-colorable is NP-hard unless r = 2 and k = 2 [1] [2].
We show that the k-colorability of an r-uniform hypergraph can be studied via an ideal of a polynomial ring which will be called the colorability ideal.
The encoding of the colorability of a hypergraph is done through several different sets of polynomials which will be defined below. Using well known theorems and properties from commutative algebra and algebraic geometry, we show that the colorability ideal of a hypergraph can be decomposed into individual coloring ideals which allow one to test if any given hypergraph is colorable by any desired color scheme.
An interesting question concerning uniform hypergraph colorings is: "What is the smallest number of edges allowed by a non-k-colorable hypergraph?" In the case of k = 2 this is known as Property B, and was introduced by F. Bernstein in [11] and studied by E. W. Miller in [12]. P. Erdős and A. Hajnal later defined m(r) to be the minimum number of edges allowed in an r-uniform hypergraph that is not 2-colorable [13]. Following Erdős and Hajnal, H. L. Abbott and D.
Hanson modified this notation to include the number of vertices in the hypergraph, n, so m n (r) is the least number of edges allowed by a non-2-colorable r-uniform hypergraph on n vertices. As it turns out, determining these values is not an easy task and so far only m(3) = m 7 (3) = 7 and m(4) are known to be tight. In 1969 Abbott and Hanson gave bounds for m n (r) with recursive type inequalities in r and n. In 1974 P.D. Seymour improved upon one of these bounds and designed a hypergraph that shows m(4) = m 11 (4) ≤ 23 [14]. Recently it has been shown by P.Östergard that m(4) = m 11 (4) = 23 proving that Seymour's construction is optimal [15].
We will show that a generalization of Abbott and Hanson's construction yields minimally non-2-colorable hypergraphs. In addition, we will provide some computational examples for upper bounds for m n (r), when r = 5, 6.

Algebraic Tools in Graph Theory
Using algebraic techniques to characterize graph theoretic properties dates back to the late 1800's when Hilbert began studying certain classes of homogeneous polynomials which turned out to be what is known as the graph polynomial. The Handbook of Combinatorics contains some excellent surveys on the uses and consequences of the graph polynomial [16].
The techniques we use to address coloring problems are part of a relatively new approach to graph theory and combinatorics. With the development of more powerful computers and more efficient algorithms, it has become possible to address combinatorial problems through these algebraic means. In 1982 D. Bayer introduced a method of determining the 3-colorability of a graph by examining systems of polynomials and applying the division algorithm [3]. Ten years later, the work of N. Alon and M. Tarsi used polynomials to prove several conjectures about the chromatic number of a graph [5]. Also, they gave equivalent conditions for a graph to be not k-colorable; we will generalize this notion to uniform hypergraphs. In 1994 L. Lovász used polynomial ideals to characterize stable sets in graphs [6].
Later, J. de Loera and C. Hillar et al. produced results concerning the algebraic characterization of a graph's colorability [7] [10]. The main tools de Loera and Hillar use in their algebraic characterizations for the colorability of a graph are polynomial ideals and Gröbner bases. Gröbner bases were introduced by B.
Buchberger in 1965 and have since become widely used in the study of polynomial ideals [17]. This thesis will generalize the above results to uniform hypergraphs, and will also utilize Gröbner bases.
The algebraic techniques developed by de Loera and Hillar et al. extended here give not only theoretical results, but also provide algorithms for solving specific problems. The process for determining the k-colorability of a hypergraph can be adjusted to detect specified color patterns required by an application. In particular, a coloring pattern known as a conflict-free coloring is addressed for k-colorings.
Conflict-free colorings were introduced in connection to work on applications to cellular networks [18] [19].
It is worth noting another line of research utilizing polynomial ideals. Many authors have utilized the rich interplay between hypergraphs and certain monomial ideals to gain insight on the structure of these ideals.
This ideal was first introduced by Villarreal [20]. By studying the associated hypergraph, many interesting results about the structure of the ideal can be achieved. This ideal has other names including the face ideal, or the facet ideal, and is also the Stanley-Reisner ideal of the appropriate simplicial complex. These ideals are not exclusive to hypergraphs; in [21] S. Jacques used tools from graph theory to study these monomial ideals.

Results
In this thesis we introduce algebraic characterizations of several different types of colorings for a uniform hypergraph H. The main result of this thesis is a generalization of a result by Hillar and Windfeldt [10].
Let the ideals J n,k , I n,k , and I G,k be defined as in [10], where K n denotes the complete graph on n vertices, that is: . (1) The graph G is not k-colorable.
(2) dim C R/I G,k = 0 as a vector space.
(3) The constant polynomial 1 belongs to the ideal I G,k .
(4) The graph polynomial P G belongs to the ideal I n,k .
(5) The graph polynomial P G belongs to the ideal J n,k .
We give a theorem that generalizes parts 1 through 4 of the above theorem to r-uniform hypergraphs. For 2-colorings we prove the following.
be the 2-colorability ideal of H and let P H,2 be the 2-color hypergraph polynomial for H. Then following are equivalent: (1) The hypergraph H is not 2-colorable.
(2) The constant 1 is an element of the ideal I 2 (H).
(4) The hypergraph polynomial P H,2 belongs to the ideal These equivalent statements rely on an ideal which completely captures the Let R = C[x 1 , . . . , x n ]. Let I(H, k) be the k-colorability ideal for H and let P H,k be the k-color hypergraph polynomial for H. Then following are equivalent: (1) The hypergraph H is not k-colorable.
(2) The constant 1 is an element of the ideal I(H, k).
(4) The hypergraph polynomial P H,k belongs to the ideal C k .
As with the 2-colorable case, Theorem 1.4 depends on the following ideal which we explore in Chapter 4 along with the k-color hypergraph polynomial. [3] D. Bayer, "The division algorithm and the Hilbert scheme," Ph.D. dissertation, Harvard University, 1982.
[15] P.Östergard, "On the minimum size of 4-uniform hypergraphs without property B," Discrete Applied Mathematics -to appear.
[16] R. L. Graham [17] B. Buchberger, "An algorithm for finding the basis elements of the residue class ring of a zero dimensional polynomial ideal," Ph.D. dissertation, Johannes Kepler University of Linz, 1965.

CHAPTER 2 Background
In this chapter we provide some additional basic definitions and develop the algebraic machinery required for the proofs in Chapters 3 and 4. Many of the definitions given here are common in combinatorics, commutative algebra, and algebraic geometry. In particular, the results given in Section 2.2 are a collection of well known facts from commutative algebra and algebraic geometry that are vital to the results in later chapters.

Graph theory and combinatorics
First we introduce some tools and important definitions from combinatorics.
The main tool we will need will be integer partitions. We will work with r-uniform hypergraphs on finite vertex sets. We start with an important example. Here, each edge is comprised of exactly three distinct vertices from the vertex set.
So, F P is a 3-uniform hypergraph on 7 vertices with 7 edges. given by:  The smallest number of colors required in a conflict-free coloring is called the conflict-free chromatic number of H, χ CF (H). The conflict-free chromatic number was introduced by G. Even et al. [1] in 2003. In Chapter 3 we will give an algorithm for finding conflict free 2-colorings of r-uniform hypergraphs. We will examine conflict free colorings of the Fano plane explicitly in Chapter 5.
To introduce the generators of the colorability ideals I 2 (H) and I(H, k) we use partitions of integers. Definition 2.2. A k-partition of a positive integer, n, is a k-length sequence of not necessarily distinct positive integers, called parts, that sum to n; For our needs, the order of the parts of any particular partition is not important. Furthermore, we wish to add some additional criteria to our partitions.
This definition is borrowed from integer compositions, which are similar to integer partitions.

Definition 2.3.
A weak k-partition of a positive integer, n, is a k-length sequence of not necessarily distinct non-negative integers, again called parts, that sum to n; (y 1 , y 2 , . . . , y k ), such that, k j=1 y j = n.

Polynomial Algebra
We will be associating each hypergraph to an ideal in a polynomial ring, since we only consider hypergraphs with finite vertex sets, we need only consider polynomial rings with a finite number of indeterminates.
For any field K and any associated polynomial ring K[x 1 , ..., x n ], Hilbert's Basis Theorem states that any ideal of K or K[x 1 , ..., x n ] will be finitely generated since K and K[x 1 , ..., x n ] are finitely generated. Throughout this thesis we will be working over the field of complex numbers since it is algebraically closed and we let R = C[x 1 , . . . , x n ].

Gröbner bases
The generalizations that we prove in this thesis require the use of a powerful tool from commutative algebra provided by B. Buchberger in [2]. The theory of Gröbner bases has become a critical part of algebraic geometry, commutative algebra, and computational and algorithmic algebra, among others. In short, a Gröbner basis is unique set of generators for an ideal.
We begin by setting an ordering on the monomials in the ring R.
Definition 2.4. A monomial ordering is a well ordering, <, on the set of monomials that satisfies mm 1 < mm 2 whenever m 1 < m 2 for monomials m, m 1 , m 2 .
Equivalently, a monomial ordering may be specified by defining a well ordering on the n-tuples α = (a 1 , ..., a n ) ∈ Z n of multidegrees of monomials Ax a 1 1 · · · x an n that With an ordering established of the ring R the leading term for any polynomial f ∈ R can be distinguished. i.e., It is also important to examine those monomials that are not the leading term of any polynomial in an ideal I.
Definition 2.8. Any monomial which is not a leading term of any polynomial in an ideal I is called a standard monomial and the set of all such monomials is denoted B < (I).
In order to utilize these important ideals, we must have a way to generate them. Thus we have the following important definition. Once a monomial ordering has been established on a polynomial ring, we can preform general polynomial division of a polynomial f by a set of polynomials {g 1 , ..., g m }. If there is a remainder r after division we write f ≡ r mod {g 1 , ..., g m }.
We introduce some notation from [3]. Let f 1 , f 2 be polynomials in C[x 1 , ..., x n ] and let M be the monic least common multiple of the monomial terms LT (f 1 ) and LT (f 2 ). Then define the difference polynomial, S(f 1 , f 2 ) to be: This notation allows us to introduce a condition for a set of generators of a polynomial ideal to be a Gröbner basis for that ideal.
Proposition 2.1 (Buchburger's Criterion, p 324, [3]). Let R = C[x 1 , . . . , x n ] and fix a monomial ordering on R. If I = (g 1 , ..., g m ) is a non-zero ideal in R, then G = {g 1 , ..., g m } is a Gröbner basis for I if and only if S(g i , g j ) ≡ 0 mod G for As stated above, reduced Gröbner bases are unique. Among many other uses, this gives us a tool for comparing ideals in a polynomial ring, and determining ideal membership. Gröbner bases and leading term ideals give a method for examining the vector space properties of quotient rings that are also C-algebras. This allows us to give bounds on the number of generators of a C-algebra, which in turn can give us information on the number of common solutions to a polynomial ideal. The following two theorems give the connection between the quotient ring of a polynomial ideal and its corresponding C-algebra.
Theorem 2.4 (Proposition 1, p227, [4]). Fix a monomial ordering, <, on R = C[x 1 , . . . , x n ] and let I be an ideal of R. Let LT (I) denote the ideal generated by the leading terms of I.
(i) Every f ∈ R is congruent modulo I to a unique polynomial r which is a C-linear combination of the monomials in the complement of LT (I) .
(ii) The elements of {x α : x α ∈ LT (I) } are "linearly independent modulo I." That is, if where the x α appearing are all in the complement of LT (I) , then c α = 0 for all α.

Square-free, radical ideals
In what follows we collect several results from algebraic geometry and commutative algebra. To this point these results have not been assembled in a single work. The use of radical, square-free generated ideals greatly simplifies the computations involved with determining the variety of the ideal in question. Since our main goal is to determine which ideals give rise to desired varieties, i.e. the variety that contains all possible proper colorings of a hypergraph, we feel this section is crucial to the results of this thesis. The majority of the results in this section come from [4] and [5], we reproduce serveral proofs for completeness only.
Let I be an ideal of the polynomial ring R = C[x 1 , . . . , x n ].
Definition 2.11. The radical of I, denoted √ I, is the set: Working with the radical of an ideal, or a radical ideal greatly simplifies computation and gives a more complete description of the structure of the ideal. It also simplifies the geometric structure associated with the ideal.
Definition 2.12. The subset of C n consisting of all of the solutions common to each polynomial in I is the variety of I, denoted V(I). Conversely, given a subset V ⊆ C n , the vanishing ideal is the set of all polynomials that vanish at every point in V , and is denoted: I(V ). The two maps V and I are related by: and are also inclusion-reversing.
The second equality above is known as The Strong Nullstellensatz, a proof is given below. Algebraic varieties are a main focus in algebraic geometry and can be quite complicated, fortunately the varieties that are associated with the ideals used to encode the colorability of a hypergraph are rather simple.
Definition 2.13. The ideal I is called zero-dimensional (as an ideal) if its variety V(I) contains only a finite number of points.
Conditions given for an ideal to be zero-dimensional are given in The Finiteness Theorem below. Working with zero-dimensional ideals of polynomial rings with coefficients coming from algebraically closed fields allows us to compare two ideals through their varieties.
The following theorem is the primary tool used to decide if the polynomials in an ideal have a common solution. It has been used extensively by De Loera, Hillar, Windfeldt and others [6], [7], [8], [9].
Theorem 2.6 (Weak Nullstellensatz). If k is an algebraically closed field and I is The contrapositive of this theorem states that if I is a proper ideal of R = Theorem 2.7 (Hilbert's Nullstellensatz). Let k be an algebraically closed field. If are such that then there exists and integer m ≥ 1 such that The following are equivalent: 2. The variety V(I) ⊂ C n is a finite set.
3. If G is a Gröbner basis for I, then for each i, Recall that such an ideal is called a zero-dimensional ideal.
Proof: Suppose I is zero-dimensional. Let G be a reduced Gröbner basis for any lexicographic ordering with x i as the smallest variable. By part 3. of the Finiteness since the ordering on I is lexicographic. So g is the non-zero polynomial required, moreover g generates the ideal I ∩ C[x i ]. Note that G being a reduced Gröbner basis and the chosen ordering being lexicographic give us that This is known as The Elimination Theorem, (theorem 3, pp 115 [4]).
Conversely, suppose I ∩ C[x i ] is non-zero for each i, and let m i be the degree of the unique monic generator of I ∩ C[x i ]. Then x i ∈ LT (I) for any monomial order, so that all monomials not in LT (I) will contain x i to a power strictly less than m i . Hence the set of monomials x α ∈ LT (I) is finite, and thus A is finite.
Definition 2.14. The square-free part of a polynomial p ∈ C[x 1 , . . . , x n ], is denoted p red and has exactly the same roots as p, but all with multiplicity 1.
Claim 2.11. If p red is the square-free part of p ∈ C[x], then p = p red .
Proof: Note that by the Strong Nullstellensatz and the definition of the squarefree part of p, p red , we have: The inclusion p red ⊇ p red follows from the definition of the radical of an ideal. Let and suppose f ∈ p red . Then f must be of the form where the set {j = 1, . . . , k } ⊆ {i = 1, . . . , k} is such that β j > 1.
The following technical lemmas illustrate the relationship between square-free polynomial generators their ideals.
where the a j are distinct. Then Proof: First we show Since p vanishes at each a j we have that For the opposite inclusion, consider the following polynomials: Note that the collection of polynomials {p 1 , . . . , p n } do not have any common zeros, hence by the Weak Nullstellensatz, there exist polynomials h j such that n j=1 h j p j = 1.
Also, we show that To show the inclusion Since g j ∈ I, and h j f j p ∈ p for all j, we have that f ∈ I + p , hence Proof: Let J = I + p 1,red , . . . , p n,red . For each i, since C is closed, we can factor p i,red into distinct factors: Thus, where the first holds since p 1,red ∈ J and the second holds by Lemma 2.12 since p 1,red has distinct roots. Repeating this argument for i = 2, . . . , n we have It follows that J is a finite intersection of maximal ideals. Since a maximal ideal is radical and an intersection of radical ideals is radical, J is radical.
It remains to see that J = √ I. The inclusion I ⊂ J is by definition of J, and the inclusion J ⊂ √ I follows from the Strong Nullstellensatz, since the square-free parts of the p i vanish at all the points of V(I). Hence and taking radicals gives Moreover, equality occurs if and only if I is a radical ideal.
Proof: Let I be a zero-dimensional ideal. By the Finiteness Theorem, V(I) is a finite set in C n , say V(I) = {p 1 , . . . , p m }. Consider the mapping which is well defined and linear. We show that φ is onto, and thus conclude that For i = 1, . . . , m, let g i be polynomials in R such that Given an arbitrary ( shows that φ is injective as well as onto. So φ is an isomorphism, which shows that Conversely, if dim C (A) = m, then φ is an isomorphism since it is an onto linear map between vector spaces of the same dimension. Hence φ is injective.
The following is a statement of a lemma of Hillar and Windfeldt. We include it here and give a proof as it is crucial for many theorems in following sections.
Lemma 2.15 (Lemma 2.1, [9]). Let I be a zero-dimensional ideal and fix a monomial ordering <. Then, Moreover, the following are equivalent: 1. I is a radical.
2. I contains a univariate square-free polynomial in each indeterminate.
Proof: The dimension condition follows from Theorem 2.14 and the fact that the standard monomials B < (I) form a basis for the algebra R/I.
Furthermore, for each i = 1, . . . , n, let p i be the unique monic generator of , and let p i,red be the square-free part of p i .
1. ⇒ 2. If I is radical then I = √ I = I + p 1,red , . . . , p n,red by Proposition 2.13, so I contains a univariate square-free polynomial in the each indeterminate.
2. ⇒ 1. By Proposition 2.13 and the fact that the univariate square-free polynomials in each indeterminate can be taken to be the square-free part of the unique monic generators of I ∩ C[x i ]. That is p i,red ∈ I for all i, so √ I = I + p 1,red , . . . , p n,red = I.
Once we can switch between a zero-dimensional radical square-free ideal and its corresponding variety, we use the ideal quotient to compute the difference of two varieties. We conclude this section with some useful results from commutative algebra and algebraic geometry. Radical ideals and their varieties behave nicely under certain operations.
Theorem 2.16 (Section 8, Chapter 4 and Proposition 16 p 188, [4]). Let I and J be ideals of R. If I and J are radical, then there is a one-to-one correspondence given by I and V such that: Moreover, Definition 2.15. The ideal quotient (or colon ideal) of the ideals I and J of R is the ideal: Ideal quotients have a nice property when restricted to ideals of polynomial rings over algebraically closed fields.
Proposition 2.17. Given two varieties V, W ⊂ C n , then We include one final theorem concerning the structure of the variety of a radical ideal.

2-Colorability Results
The approach to the colorability problems we use involves translating properties of a combinatorial object, i.e. a hypergraph, into the language of commutative algebra, namely into ideals and varieties of polynomial rings. To address the encoding in the 2-colorability case, we will define a system of polynomials that will capture the colorability of the hypergraph H. In addition we will provide polynomials that capture individual coloring schemes, and show how they relate to the overall colorability of the hypergraph. Here we collect all of the results concerning 2-coloring in this chapter, we provide proofs in Section 3.2.
We define a 2-coloring as a map c : V (H) → {−1, 1}, and note that this formulation is equivalent to the definition above. We introduce some notation that will allow us to define our polynomials and ideals.
These polynomials are crucial in the definition of the 2-colorability ideal for H, I 2 (H): As the name implies, the ideal I  As an analogue to the commonly used graph polynomial, define the hypergraph polynomial for 2-colorability, P 2 (H), by: where e j ⊂ [n], |e j | = r, is an edge in H. A similar generalization of the graph polynomial was introduced in [1]. We state a theorem that captures the generalization for 2-colorability here and postpone the proof to the next section. We can now state our main result for 2-coloring r-uniform hypergraphs. It is a generalization of Hillar and Windfeldt's Theorem 2.1 in [2]. (1) The hypergraph H is not 2-colorable.
(2) The constant 1 is an element of the ideal I 2 (H).
(4) The hypergraph polynomial P 2 (H) belongs to the ideal  The 2-coloring scheme ideal, J 2 (U ), is the ideal that encodes the colorability of H by the edge signatures in U : x e,i − a : e ∈ E(H) .
These ideals play an important role in distinguishing one proper coloring pattern from another.

2-Colorability Proofs
In this section we let A be the set of all proper edge signatures for a 2-coloring of H. We first establish the ideal characterization of a proper 2-coloring of H by proving that our f e polynomials encode proper edge coloring. Then we have that: Thus at least one of the factors, (c j x e,1 + x e,2 + · · · + x e,r )(x e,1 + c j x e,2 + · · · + x e,r ) · · · (x e,1 + x e,2 + · · · + c j x e,r ) of f e will be zero. Hence f e (c) = 0.
(⇒) Assume f e (c) = 0. Assume e is not properly colored. Then each factor in: (c j x e,1 + x e,2 + · · · + x e,r )(x e,1 + c j x e,2 + · · · + x e,r ) · · · (x e,1 + x e,2 + · · · + c j x e,r ) has sum either, Remark 1. A note on Theorem 1.3: in the r-even case, we need not cycle the c j coefficient through r factors of f e when c j = r 1 − r 2 + 1 = 1 since all coefficients will be 1. Thus, in the even case, f e will contain one factor of the form (x e,1 + x e,2 , · · · + x e,r ) and the remaining (r − 1) factors where c j = 1 are omitted. This does not change the variety V(I 2 (H)), however it does simplify some computations within I 2 (H).
We now note how the hypergraph polynomial encodes a generalization of the graph polynomial.

Proof: (Theorem 3.2)
We note that H is not properly 2-colored if and only if there exists an edge with all vertices assigned the same color. This happens if and only if that edge has a vertex sum equal to r times the value of a single color. Since our colors have been restricted to ±1, the following are equivalent: • H is not properly 2-colored.
• ∃ an edge in E(H) with vertices colored by all 1's or all -1's.
• ∃ an edge whose vertices sum to ±r.
So, given a 2-coloring of H with the colors ±1: iff either Since x i = ±1, (2) happens iff ∃ e j ∈ E(H) that has vertices colored by either all 1's or all -1's, that is, The following lemmas and their proofs are analogues of Lemmas 3.1 and 3.4 in [2].  Proof: These statements follow from the fact that the ideals I 2 (H), and x 2 i − 1 : i ∈ V (H) + P 2 (H) are radical and from Lemma 3.7.
Then since V (I 2 (H)) = ∅, and both I 2 (H) and x 2 i − 1 : i ∈ V (H) are radical, the ideal quotient: Also, we have that So χ H (2) = 0, and thus H is not 2-colorable.
Proof: (Theorem 3.4) Let U ⊆ A, U non-empty. Consider the ideal: x e,j − a : e ∈ E(H) .
From the first set of polynomials we see that any common solution will be an n-tuple of 1's and -1's. Also, it is clear that for every edge, e ∈ E(H): x e,j = a, for some a ∈ U.
Since each factor is the sum of the values of the vertices in the edge e, this can happen if and only if the edge is colored by a signature in U .
Theorem 3.4 also gives us the following as a corollary. This is one of the key components to the proof of the decomposition Theorem 3.5. Proof: This follows from Theorems 1.3 and 3.4.
We can now prove our main decomposition theorem for 2-colorability.
Proof: (Theorem 3.5) Since the ideals I 2 (H) and J 2 (U ) contain square-free univariate polynomials in each indeterminate, they are radical. Also, since by Theorem 2.8, we have that: As a first corollary to Theorem 3.5, we have that given some U ⊆ A, we can test to see if I 2 (H) can be colored by the edge colors/signatures in U .
where a 1 , a 2 ∈ A are proper edge signatures and a 1 = −a 2 .
Let a 1 and a 2 represent the two permutations of the colors on the vertex set. Then by Theorem 3.5, Moreover, since the two edge signatures a 1 and a 2 are permutations of each other, we have that a 1 = −a 2 .
(⇐) Assume that, where a 1 , a 2 ∈ A are proper edge signatures and a 1 = −a 2 .
Since a 1 = −a 2 , we have that the signatures a 1 and a 2 are permutations of the colors assigned to the vertices. We also have that |V(I 2 (H))| = 2 by Theorem 3.5.

Conflict-free coloring
Our goal is to show how we can recognize hypergraphs with χ CF (H) = 2 and  See Chapter 5 for a detailed illustration of Theorem 3.12.

CHAPTER 4 k-Colorability
In this chapter we introduce our complete generalization of Hillar and Windfeldt's Theorem 1.1, along with the associated k-colorability results. We provide all statements of the k-colorability results in Section 4.1 and collect their proofs in Section 4.2. We also state and prove some theoretical results concerning list colorings of uniform hypergraphs and provide an algebraic characterization of the t-choosability of a uniform hypergraph. We end the chapter with an algebraic consequence of the structure of the ideals defined in this thesis.

k-Colorability Results
To address the k-colorability of an r-uniform hypergraph, we use similar techniques as in the 2-colorable case. Let r ≥ 2 be an integer. Let H be an r-uniform we cannot directly generalize this to the k-colorable case using primitive k th roots of unity as was done in [1] and [2] for graph colorings. Instead, for the general k-colorability we will utilize prime numbers as our colors.
Let k ≥ 2 be an integer, and let P k be the set of the first k primes. Define a x e i .
A proper coloring does not allow a monochromatic edge, thus we encode all of the non-monochromatic edges and use them to force a proper coloring.
Define the following as the set of proper edge products: Note that for any color p ∈ P k , The k-tuples of exponents, (α 1 , . . . , α k ), of the prime products in A are precisely the set of all proper k-integer partitions of r.
Consider the ideal:  x e i − a .
Define I(H, k) as: We can now restate Theorem 5 from Chapter 1 which defines the k-colorability ideal for H. Define the hypergraph polynomial for k-colorability, P H,k , by: (1) The hypergraph H is not k-colorable.
(2) The constant 1 is an element of the ideal I(H, k).
(4) The hypergraph polynomial P H,k belongs to the ideal C k .
Moreover, by the structure of the ideal I(H, k), we can classify certain colorings of H. Let U be a non-empty subset of A.
Definition 4.1. The k-coloring scheme for the r-uniform hypergraph H given by U is the set of all colorings of H with the edge products in U .
We define the k-coloring scheme ideal, J(U, k), as the ideal that encodes the colorability of H by the edge products in U : x e i − a : e ∈ E(H) .   As an illustration of Corollary 4.8, let U be the subset of all proper edge products that correspond to the particular k-integer partition of r, α = (α 1 , . . . , α k ).
That is, where the p 1 , . . . , p k are permuted in all possible ways. Then the variety of the k-coloring scheme ideal J(H, U ) contains the proper colorings of H in which α j vertices share the same color in each edge, for j = 1, 2, . . . , k. Hence each edge contains the same color pattern associated with the partition α, although the colors assigned to each part α j may differ on distinct edges.

k-Colorability Proofs
In this section we let A be the set of all proper edge products for a k-coloring of H. Also let P k be the set of the first k prime numbers.
Proof: (Claim 4.1) For any i ∈ V (H) the associated polynomial in C k . The edge product of e is then p r t , thus c is a common solution of P H,k + C k .
(⇐) Assume P H,k + C k has a common solution, c. Since C k vanishes at c, each vertex is assigned a single power of a prime in P k . Also, since P H,k (c) = 0 there exists an edge e ∈ E(H) and a prime p t ∈ P k such that: Thus the edge e has an edge product of p r t and is not properly k−colored.
Proof: (Theorem 4.6) Let U be a non-empty subset of A. (⇒) Assume c is a common solution to J(U, k). By the definition of C k , c is a k−coloring of H.
Moreover, since the product of the vertices in each edge is a value in U , H is properly colored by the edge products in U .
(⇐) Assume H is colorable by the edge products in U . Let c be any such k−coloring. Then c assigns an edge product from U to each edge, thus c is a solution to J(U, k).
Proof: (Theorem 4.7) Note that since C k is contained in both I(H, k) and J(U, k), and C k contains univariate squarefree polynomials in each indeterminate, C k , and thus I(H, k) and J(U, k), are all radical by Theorem 2.15. Since, we have that, Moreover, since I(H, k) and J(U, k) are radical, we have that: by Theorem 2.8.
Proof: (Corollary 4.8) Let U be a non-empty subset of A.

By Theorems 3.4 and 4.7, H can be colored by the edge products in U if and only
if V(J(U, k)) ⊆ V(I (H, k)). Since both I(H, k) and J(U, k) are radical: V(J(U, k)) ⊆ V(I (H, k)) if and only if I(H, k) ⊆ J(U, k).

Conflict-free k-Colorings
In this section we address conflict-free colorings of uniform hypergraphs when using possibly more than two colors. As with conflict-free 2-colorings, a conflictfree k-coloring is a proper coloring in which each edge contains a vertex whose color is not repeated by any other vertex in the edge. We capture this coloring with the following edge products. Let U CF be the subset of proper edge products, A, such that: where the k-tuple (α 1 , . . . , α k ) is a proper k-integer partition of r. Define U CF to be the set of all conflict-free edge products. We can determine if a uniform hypergraph contains a conflict-free k-coloring via the following theorem. Proof: Let U = U CF , the result follows from Corollary 4.8.

List Colorings
In this section we introduce list colorings of uniform hypergraphs and provide an algebraic characterization of the t-choosability of a uniform hypergraph. We note that the characterization holds for graphs as well. As before we collect all results in Section 4.4 and postpone the proofs until Section 4.5.
Let H be an r-uniform hypergraph on the vertex set [n]. Let P k be the set of the first k primes. For each vertex v ∈ V (H) let be a given list of colors where each p v i ∈ P k . then H is t-list choosable.
As in the k-Colorability section we define the following set as the proper edge products for an r-uniform hypergraph: We only restrict the possible colors to be the first k primes for computational considerations; the colors need only be distinct relatively prime elements from a unique factorization domain contained within an algebraically closed field of coefficients.
Let S = {S v } v∈V (H) be a collection of lists of colors and consider the following ideals:  The corresponding variety is the collection of the appropriate list colorings.

List Coloring Proofs
Proof: (Proposition 4.10) (⇒) Let v be any vertex in H. Assume the v th Then the v th coordinate of c has a value p v ∈ S v . Thus the vertex v is colored by the list S v .
(⇐) Assume c ∈ C n is a coloring of H in which the vertex v is colored by a member of the list S v = {p v 1 , . . . , p vt }. Then the v th coordinate of c has a value p v i ∈ S v . Hence exactly one factor of the product is zero, so c is a solution of the ideal C v .
Proof: (Theorem 4.11) Since the argument in Proposition 4.10 holds for every v ∈ V (H), i.e. for each indeterminate x i ∈ C[x 1 , . . . , x n ], and

Ideal Primary Decompositions
In this section we examine the relationship between the collection of proper colorings of a uniform hypergraph and the structure of the ideals that encode these colorings. The encodings for colorability given in Chapters 3 and 4 owe much of their ease to the structure of their ideals. Each colorability ideal is constructed so that its variety is the collection of all possible colorings that satisfy some condition.
As a result, each variety is a finite collection of points in C n . Moreover, we can make a statement about the multiplicity of these solutions.   4.16 (Propositions 9 and 10,[3]). I c is maximal and Prime ideals are important in determining the structure of an ideal. If an ideal I can be uniquely written as an intersection of distinct prime ideals Q i then the intersection is called the minimal primary decomposition of I: Similarly, if a variety V can be written as union of disjoint irreducible varieties V j the union is called the minimal decomposition of V : For more information on primary decompositions of ideals and decompositions of varieties, see [3] or [4]. List of References CHAPTER 5

Computation
In this chapter we provide some detailed examples of applications of the theorems in Chapters 3 and 4. In the first section we address the coloring scheme ideals and conflict-free colorings of the Fano plane. In Section 5.2 we provide a technique for determining if a uniform hypergraph can be properly colored which we utilize in Section 5.3. In Section 5.3 we give a generalization of the construction given by Abbott and Hanson in [1], and improved on by Seymour in [2]. We end the chapter by providing some computations on the chromatic number of Stable Kneser hypergraphs.

2-colorability
The corresponding 2-colorability ideal I 2 (F P ) given by Theorem 1.3 is: Which gives us the coloring scheme ideals: Using a Gröbner basis package in a computer algebra system like Mathematica or Singular we can show that all of the above ideals contain the constant 1 and thus by Theorems 1.3 and 3.5, the Fano plane is not 2-colorable.

Conflict Free Colorings of the Fano Plane
To further demonstrate Theorem 3.5 and give an example of a conflict-free coloring we consider the Fano plane with an edge removed. Note that, Let the Modified Fano plane be the hypergraph F P = F P \ {1, 2, 5} where: 2, 3, 4, 5, 6, 7} and The corresponding 2-colorability ideal is: The reduced Gröbner basis for I 2 (F P ) with respect to the monomial ordering This also tells us that χ CR (F P ) = 2.
Next we examine the coloring scheme ideals for F P : The reduced Gröbner bases for J 2 ({1}), J 2 ({−1}), and J 2 ({1, −1}), with respect to the monomial ordering x 1 > x 2 > · · · > x 7 , are: Moreover, we see that the Gröbner bases for I 2 (F P ) and J 2 ({1, −1}) with respect to the monomial ordering x 1 > x 2 > · · · > x 7 , are equal. In addition, the 1}) is the same also. Hence the 2-colorability ideal and the intersection of the color scheme ideals are identical and we conclude that the modified Fano plane F P is 2-colorable, and also admits a conflict-free 2-coloring. Moreover, it can be similarly shown that the Fano plane with any single edge removed is properly 2-colorable and also has a conflict-free coloring.

Color Extensions
The coloring ideals given in Theorems 1 has a common solution.
Proof: Without loss of generality, assume only x 1 is colored, we can iteratively apply the results if more vertices are colored. Also, assume that the color chosen is appropriate: ±1 for I 2 (H) or some p ∈ P k for I(H, k).
Since appropriate colors are used, either the polynomial x 2 1 − 1 in I 2 (H) or p∈P k (x 1 − p) in I(H, k) will vanish. Moreover the f e or f e,k polynomials in either ideal will have each incidence of x 1 replaced with the chosen color. This will require any common solution of either ideal to contain the chosen color in the first coordinate. Computing a Gröbner basis for the remaining polynomials determine if they have a common solution with the first coordinate fixed. Thus any common solution will be an extension of the partial coloring.
This technique allows us to quickly determine that a hypergraph is properly colorable withour knowing the complete Gröbner basis of the colorability ideal. If a partial coloring does not extend to a proper coloring, we cannot conclude that the hypergraph is not colorable, as a different partial coloring may extend to a proper coloring. Extending a partial coloring can be used to test for non-colorability, as long as all possible initial colorings are tested on the chosen vertices. Depending on the number of vertices in H and the uniformity of H, coloring as few as 3 vertices can improve computing time.

Constructions
Let H be an r-uniform hypergraph on n vertices. We defined m n (r) be the least positive integer m such that: |E(H)| = m, and H does not have Property B. That is, m n (r) is the least number of edges in a non-2-colorable, r-uniform hypergraph on n vertices. Abbott and Hanson give (among others) the following inequalities for m n (r) in [1]: When r = 4 and n = 3, Seymour gives a construction which improves on the bound given by the inequality above, [2]. For the r odd case, a generalized version of Seymour's construction cannot improve on the bounds given by Abbott and Hanson in [1]. When r is even, however, we show that a generalization of Seymour's construction can improve these bounds. We specifically show the r = 6 construction and improve the bounds on m 23 (6) to 180.

Optimizing Seymour's Construction.
Let n and r be positive integers. The construction given in [2] generalizes to the following: Take S = [2r + n] = {1, 2, . . . , 2r + n}, and let A be a non-2-colorable hypergraph on {2r + 1, 2r + 2, . . . , 2r + n} with m n (r − 2) edges. Define, Let E be a subset of D such that the following two conditions hold: (i) If X, Y ∈ D and X ∪ Y = [2r + n], then either X or Y is a member of E.
(ii) If |Q| = r − 1, and Q is a subset of any member of D, then Q is a subset of a member of E.
Claim 5.2. If F = C ∪ E is an r-uniform hypergraph on 2r + n edges, then F is not 2-colorable.
Proof: Fix n and r in Z + . Let F = C ∪ E.
Suppose that F is 2-colorable, or has property B; that is, suppose Z is a subset of S that intersects every member of F , but contains no member of F . Then S \ Z is also a set intersecting each member of F that contains none, so we may assume that, This follows from counting the maximum number of vertices in each edge that can be in either Z or S \ Z. Since A is not 2-colorable, (1) Since Y ⊂ Z, and {1, 2} ⊂ Z, X ⊆ Z. This contradicts the hypothesis that Z contains no member of F , thus F does not have Property B.
Next we introduce the generalized cube graph, GQ r as follows. First define the following set, let J be the set of all r − 1 element subsets of any element of D. Let GQ r be a graph with vertex set V (GQ r ) = D, so each member of D is assigned a vertex in GQ r . The edge set for GQ r is defined as follows: Defined this way, the vertex, v X , for any given element in X ∈ D will be adjacent to the complement of X in [2r], and all vertices associated with elements in D who share an r − 1 element subset with D.
The graph GQ r is r + 1-regular hypercube on 2 r vertices with all antipodal diagonals as edges. A vertex cover, V C (G), of a graph is a subset of the vertex set of G such that every edge in E(G) is incident to at least one vertex in V C (G).
The following claim shows that there is a correspondence between the set E and V C (GQ r ).
Claim 5.3. Let V C be a vertex cover of G. The members of D that correspond to vertices in V C satisfy conditions (i) and (ii).
Since for all X ∈ D the edge (v X , v X c ) is incident to at least one vertex in V C , condition (i) is satisfied.
Since every edge in E(G) is incident to a vertex in V C , every r − 1 element subset of any member of D is represented by at least one vertex in V C , satisfying condition (ii).
Claim 5.3 asserts that any vertex cover will work in constructing a non-2colorable hypergraph. By adding the members of D that are present in a vertex cover of G to C we will create a hypergraph that satisfies Claim 5.2. This leads us to utilize a minimal vertex cover. Adding a minimal vertex cover of GQ r to C will create a non-2-colorable r-uniform hypergraph with the minimal number of edges allowed by Seymour's construction. We find minimal vertex covers of GQ 5 and GQ 6 using linear programming in Sections 5.2.2 and 5.2.3. We note that while this will improve on Abbott and Hanson's upper bound, it may not be the best possible construction. However, m(6) ≤ 180 is the best known upper bound.

5-Uniform Construction
We can show minimality for the 5-uniform case, we begin with the 3uniform, non-2-colorable hypergraph on 7 vertices, the Fano plane. Let F P * = (V (F P * ), E(F P * )) be the following: Note that this is the same hypergraph as above, with renamed vertices. We construct a 5-uniform hypergraph, following Abbott and Hanson in [1], as follows: we let Let E ⊂ D satisfy conditions (i) and (ii) above and let F 5 = C ∪E. To choose E, we utilize the minimal vertex cover technique given in Claim 5.3. The set F 5 is given explicitly in Appendix 1.
Further, it can be shown computationally, by using the color extension technique described in Section 5.2, that removing any edge from F 5 will yield a 2colorable hypergraph. That is, F 5 is critical.
Further, for any r odd, GQ r is a bipartite graph and, hence, the minimal vertex cover for GQ r has size 2 r−1 . This agrees with the upper bound provided by Abbott and Hanson in [1].
It can be shown, again using the computation of color extensions, that removing any edge from F 6 will yield a 2-colorable hypergraph, hence F 6 is critical.
Moreover, this construction improves on Abbott and Hanson's upper bound. Using the inequalities given at the beginning of this chapter, Abbott and Hanson guarantee a non-2-colorable 6-uniform hypergraph on 23 vertices with 196 edges. We have improved this to 180. In addition, Claim 5.3 implies that this is an optimal construction possible using conditions (i) and (ii).

The Chromatic Number of Stable Kneser Hypergraphs
As a further application of our method in this section we provide computation of chromatic numbers for some Stable Kneser hypergraphs. Let r, n ≥ 1 be positive that is, any two elements of S are at least a 'distance of r apart' modulo n. For r ≥ 2, k ≥ 2, the r-stable k-uniform Kneser hypergraph, KG r [n] k r−stab is the hypergraph with vertex set consisting of all r-stable k-element subsets of [n].
The edge set is formed by r-tuples S 1 , . . . , S r of pairwise disjoint vertices, i.e. of pairwise disjoint r-stable k-element subsets of [n].
Stable Kneser hypergraphs generalize to Kneser hypergraphs introduced by M. Kneser in 1955, [3]. In 1978 Lovász proved Kneser's conjecture on the chromatic number of Kneser graphs, [4]. Later Alon, Frankl, and Lovász proved a conjecture of Erdős on the chromatic number of Kneser hypergraphs KG r [n] k , [5]. In [6], the authors conjecture that the chromatic numbers of Stable Kneser hypergraphs are the same as the chromatic numbers of Kneser hypergraphs. Conjecture 5.4. Let n, k, r > 0 be integers such that n ≥ rk. Then χ KG r [n] k r−stab = n − (k − 1)r r − 1 .
The conjecture is known to hold only for r being a power of 2, [6]. Recently, F. Meunier supported the conjecture by computation. We extend Meunier's computation using the methods developed in this thesis. These computations are done using partial color extensions described in the previous section. We can conclude that the conjecture holds for: r = 3, k = 4, n ≤ 15 k = 5, n ≤ 18 k = 6, n ≤ 21 k = 7, n ≤ 24 r = 5, k = 2, n ≤ 14.
We note that the Stable Kneser hypergraph KG r [n] k r−stab with r = 3, k = 7, and n = 24 has 288 vertices and 9568 edges.

CHAPTER 6 Conclusions
The techniques developed in this thesis extend the algebraic methods developed and implemented by authors such as Alon, De Loera, Hillar, and Lovász. Coloring of hypergraphs is the first application of these techniques. The results in this thesis provide new results on colorings of uniform hypergraph using polynomial ideals. The first possibility for generalization is to remove the uniformity condition.
Open Problem 1. Extend the results of this thesis to the non-uniform case.
In addition, we believe that these techniques can also extend computation. In Chapter 1 we mentioned the question posed by Miller, "What is the least number of edges allowed in a non-2-colorable uniform hypergraph?" Using Theorem 1.5 we can address a generalization of Miller's question. We set m n (r, k) to be the minimum number of edges allowed in a non-k-colorable r-uniform hypergraph on n vertices. Let M n (r, k) be the set of all positive integers m such that there exists an r-uniform hypergraph on n vertices, H, that is critically non-k-colorable and where |E(H)| = m. Note that this set is related to the notation introduced by Erdős and Hajnal: m(r) = min n M n (r, 2).
As an application of Theorem 1.5 we wish to examine these sets.
Open Problem 2. Given positive integers r, k, and n, examine the structure of the set M n (r, k).
In addition to coloring, using square-free generated radical ideals has potential for detecting certain subgraphs and subhypergraphs. It would be beneficial to find a polynomial ideal associated with a given graph or hypergraph which encodes the structure of the induced subgraphs present. Does there exist an ideal that satisfies the following question: Open Problem 3. Given hypergraphs H, and G. Find an ideal I which contains polynomials that have a common solution if and only if the hypergraph H contains G as an induced subhypergraph.
The ideals constructed in theorems like Theorem 1.3 and 1.5 have an explicit structure that was designed to fit a specific need. Other ideals such as the Stanley-Reisner ideal for simplicial complices and edge ideals provide a more general interpretation of the structure of a graph or hypergraph. We propose further study to better understand the connection between properties of hypergraphs, the associated ideals, and their corresponding varieties.