PROPERTIES of DEGREE SEQUENCES of K-UNIFORM HYPERGRAPHS

PROPERTIES OF DEGREE SEQUENCES OF k-UNIFORM HYPERGRAPHS

NEGARSADAT MIRSATTARI

Contents Dedication 2 1. Introduction 3 1.1. Basic Deﬁnitions 4 1.2. The Characterization Problem for Hypergraphic Sequences 4 1.3. ThePolytopeofHypergraphicSequences 6 2. CharacterizationsofDegreeSequences 8 2.1. Characterizations of Graphic Sequences 8 2.2. DegreeSequencesofOtherClassesofGraphs 10 2.3. PropertiesofHypergraphicSequences 12 3. ASetofNecessaryConditionsonHypergraphicSequences 14 3.1. AnOrderonClassesofHyperedges 15 3.2. Upperbounds and Proofs 18 3.3. More on Upperbounds 21 4. Polytope of Degree Sequences 24 4.1. Properties for Dn 24 4.2. PolytopeofHypergraphicSequences 28 4.3. Generalizing Koren Properties for Dn(k) 31 5. Reduction to the Problem of Lattice Points 35 5.1. The Lift Operation and Hidden Points 35 5.2. A Bijection and Reduction to Lattice points 37 5.3. Some Interesting Properties of the Threshold Hypergraphs 40 Appendix A. Preliminary Steps for Theorem 3.8 44 References 52

1 2 February 9, 2011

Dedication. To Navid, my brother, who proves me ∄ l ∈{limits} one can not rise beyond. PROPERTIES OF DEGREE SEQUENCES OF k-UNIFORM HYPERGRAPHS 3

1. Introduction Problems involving graphs first appeared in the mathematical folklore as puz- zles [1]. Since then, graphs have been useful tools in other fields of mathematics, such as algebra, number theory, geometry, topology, and geometry [2], as well as in other fields of computer science, such as scheduling, optimization, networks, and bioinformatics. Graphs have been also extensively used in other fields of science, such as electrical engineering, chemistry, psychology, and economics [1].

Using graphs is a powerful way to model systems in which relations involve a pair of elements. However, for more complicated systems, where relations involve sets of elements of size larger than two, we need concepts more general than the tra- ditional graphs. A generalized form of graphs which can model relations involving sets of elements, are called hypergraphs. In particular, a special case of hypergraphs, where all of the edges are distinct sets and of the same size k, are called k-families or k-(uniform) hypergraphs. k-hypergraphs have been the main focus of studies in the hypergraph theory.

Hypergraphs can be used to simplify the already existing graph models, as well as to model more complicated systems. Hypergraphs also facilitate generalizing concepts or systems modeled by graphs. Modeling a system, usually we are interested in more information than just individual states of the system.

The degree sequence of a graph or hypergraph is the integer sequence which records the number of collections a given element is contained in. Degree sequences of (hyper)graphs and convex hull polytopes of the degree sequences provide us with extremely helpful tools to model and gather information about possible states of complicated systems, boundary or extreme possible states, transitions between different states, and a wide range of statistics about the states of the systems.

In this paper, we investigate properties of degree sequences of k-hypergraphs as well as the properties of their convex hull polytopes. No characterization of these sequences is known. In Section 3, we introduce a set of necessary conditions on a n n sequence ϕ = (d1, d2, .., dn) ∈ N , where i=1 di ≡ 0 (mod k)tobe k-hypergraphic, as follows (Theorem 3.8): P z z dj ≤ αiMi + dj − γiMi /γz+1 αz+1 j∈S i=0 j∈T i=0 X X X X for all S,T such that S ∪ T ⊂ [n] and S ∩ T = ∅.

The Mi are parameters measuring the maximum possible number of hyperedges in the class of hyperedges containing αi vertices from S and γi vertices from T . We also introduce conditions similar or equivalent to this result that we use in proving properties of the convex hull polytopes of k-hypergraphic sequences.

For properties of the convex hull polytopes of k-hypergraphic sequences, ﬁrst in Section 4.3, we generalize some known properties of the graphical case to hypergraphic case, using our result for properties of degree sequences of k-hypergraphs. Then in Section 5, we reduce an open problem concerning holes in the polytope 4 February 9, 2011 of degree sequences (Problem 1.12) to a problem on lattice points (Problem 5.9.) Also we use this reduction to better understand these holes if they exist.

In the rest of this section, we will explain the basic deﬁnitions and the problems under study. In Section 2, we review known characterizations or properties of several classes of graphs. In Section 3, we explain our results about the properties of the degree sequences of hypergraphs. Appendix A contains the preliminary steps to this result. The known properties and our result about polytopes of degree sequences of hypergraphs form Section 4. In Section 5, we reduce a problem concerning the polytope of degree sequences to a problem on lattice points and use this reduction to gain some interesting properties of the polytope of degree sequences. 1.1. Basic Deﬁnitions. A hypergraph is a generalization of a graph, where an edge can connect any number of vertices. Formally, a hypergraph H is a pair H = (V, E), where V is a set of elements, called vertices or nodes, and E = {S1,S2, ..., Sm}, where each Si is a non-empty subset of V , called a hyperedge, block or link. A special case of hypergraphs, where all of the Si are distinct and of the same size k, are called k-families or (simple) k-(uniform) hypergraphs. A simple graph (without loops and multiple edges) is a 2-family.

Deﬁnition 1.1. (k-families or k-uniform hypergraph:) A “k-uniform hypergraph” K is a pair K = (V, E), where V is a set of elements and E = {S1,S2, ..., Sm} is a collection of distinct k-subsets of V .

In this text, for the purpose of simplicity we consider V = {1, 2, ..., n} = [n] and [n] therefore, for all i ∈ [m], Si ⊂ [n]. In particular, for k-families Si ∈ k . The degree of a vertex v, denoted by deg(v), is equal to the number of (hyper)edges incident with the vertex v. The (ordered) degree sequence of a (hyper)graph with vertices v1, v2, ..., vn is deﬁned as d(H) = (deg(v1), deg(v2), ..., deg(vn)). Note that in this deﬁnition vertices are ordered and there is no non- increasing condition on the sequence. Also note that the domain of a degree is the natural numbers, i.e. N = {0, 1, 2, 3,...}. One can also look at the degree sequences as a n dimensional point or vector in Nn, where the value of the ith dimension is equal to deg(vi).

Example 1.2. H = ([6], E), where E = {{v1, v2, v3}, {v1, v2, v5}, {v1, v3, v6}, {v4, v5, v6}}, is a 3-family on [6]. Here, we will simply write E = {123, 125, 136, 456}. d(H) = (3, 2, 2, 1, 2, 2) is the degree sequence of H. Figure 1 shows a graphic representation of H.

1.2. The Characterization Problem for Hypergraphic Sequences. A nonnegative integer sequence d = (d1, d2, .., dn) is called graphic if there exists a (simple) graph G, for which d is the degree sequence. In this case, d is called realizable by G and G realizes or is a realization of d. A graphic sequence might be realizable by more than one graph. When a sequence is realized only by one graph (or its isomorphisms) it is called uniquely realizable. Similarly, PROPERTIES OF DEGREE SEQUENCES OF k-UNIFORM HYPERGRAPHS 5

Figure 1. Graphic representation of the 3-family from Example 1.2

Deﬁnition 1.3. (k-hypergraphic sequence:) A non-negative integer sequence d = (d1, d2, .., dn) is called “k-hypergraphic” if there exists a k-family H, for which d is the degree sequence.

Similarly, the terms realizable, to realize, realization, and uniquely realizable could be used for hypergraphic sequences.

Example 1.4. The sequence d = (3, 2, 2, 1, 2, 2) from Example 1.2 is the degree sequence for several 3-families , but it is not 6-graphic.

How could we recognize if a given integer sequence d = (d1, d2, ..., dn) is k- n hypergraphic? It is easy to see that i=1 dn ≡ 0 (mod k), but this is not suﬃcient. P Open problem 1.5. Give a characterization of k-hypergraphic sequences, d = (d1, d2, .., dn).

This problem has been studied extensively. In the literature, characterizations for graphic sequences have been proven in diﬀerent formats. Also the characterizations of some speciﬁc classes of graphs including bipartite graphs and r-graphs are know. However, in the general case of hypergraphs, the answer to this problem is not known.

In Section 2, we review some known characterizations and properties of several classes of graphs, including simple graphs, r-graphs and hyper graphs. In Section n 3, we introduce a set of necessary conditions on a sequence ϕ = (d1, d2, .., dn) ∈ N , n where i=1 di ≡ 0 (mod k) to be k-hypergraphic, as follows (Theorem 3.8): z z P dj ≤ αiMi + dj − γiMi /γz+1 αz+1 j∈S i=0 j∈T i=0 X X X X for all S,T such that S ∪ T ⊂ [n] and S ∩ T = ∅.

The Mi are parameters measuring the maximum possible number of hyperedges in the class of hyperedges containing αi vertices from S and γi vertices from T . We 6 February 9, 2011 also introduce propositions containing conditions similar or equivalent to this result that we use in proving properties of the convex hull polytopes of k-hypergraphic sequences.

1.3. The Polytope of Hypergraphic Sequences. Most well-known characterizations of graphic sequences (see Section 2.1) are systems of (redundant) inequalities. Each system could be used to determine a polytope in Rn. These polytopes and their properties, such as extreme points and adjacency of points, have been the subject of several studies. The construction of such polytopes has also been extended to k-hypergraphic sequences.

Deﬁnition 1.6. Deﬁne the set of all k-families on a given set [n] as Fn(k)= {G : G is a k-family on [n]}.

Deﬁnition 1.7. Deﬁne the set of integer points of all K-families in Nn as DSn(k)= {(d1, d2, ..., dn)|(d1, d2, ..., dn) is k-hypergraphic }, i.e. DSn(k)= { degree sequences of members of Fn(k)}.

n Deﬁnition 1.8. Deﬁne the polytope of k-hypergraphic sequences Dn(k) ⊂ R as Dn(k)= Convex hull of {DSn(k)}.

For the special case k = 2, the K is dropped and we write, Fn, DSn, and Dn.

Deﬁnition 1.9. (integer point:) Call each point d = (d1, d2, ..., dn) ∈ Dn(k) an “integer point” of Dn(k) if

(1) for all i ∈ [n], di ∈ N and n (2) i=1 di ≡ 0 (mod k). P Each K ∈ Fn(k) maps to an integer point d(K) in Dn(k), which is the vector representation of K’s degree sequence. We know that it is possible for an integer point in Dn(k) to be the degree sequence for more than one hypergraph in Fn(k). But is each integer point in Dn(k) the degree sequence for at least one hypergraph in Fn(k)?

Example 1.10. The degree sequence of the 3-family K from Example 1.2, d(k)= (3, 2, 2, 1, 2, 2), is an integer point in D6(3). This sequence is the degree sequence for several other 3-families on [6], including K′ = ([6], E′), where E′ = {126, 125, 136, 345}. All these 3-families on [6] are members of F6(3).

Deﬁnition 1.11. We call an integer point d ∈ Dn(k) a “hole” if there does not exist a k-family on [n], for which d is the degree sequence. i.e. n n d = (d1, d2, ..., dn) ∈ Dn(k) is a “hole” if d ∈ N and i=1 di ≡ 0 (mod k) , but d∈ / DSn(k). P PROPERTIES OF DEGREE SEQUENCES OF k-UNIFORM HYPERGRAPHS 7

Open problem 1.12. Are there any holes in Dn(k)?

For the case k = 2, it is known that there is no hole in Dn. However this problem is still open for cases k > 2. We will review the known properties about Dn and Dn(k) in Section 4. In particular, in Section 4.3, we generalize some known properties of Dn to Dn(k), using our result for properties of degree sequences of k-hypergraphs. Then in Section 5, we reduce an open problem concerning holes in the polytope of degree sequences (Problem 1.12) to a problem on lattice points (Problem 5.9.) Also we use this reduction to prove some interesting properties of the polytopes of degree sequences. 8 February 9, 2011

2. Characterizations of Degree Sequences The problem of characterizing degree sequences of k-uniform hypergraphs, i.e. Problem 1.5, is a long standing open question in hypergraph theory [2]. In the literature, characterizations for graphic sequences have been proved in diﬀerent formats. Also the characterizations of some speciﬁc classes of graphs including bipartite graphs and r-graphs are known. However, in the general case, the answer to this problem is not known.

In the rest of this section, ﬁrst we mention some of the well-known characterizations of graphic sequence in Section 2.1 and some of the known characterizations or properties of other classes of graphs in Section 2.2. Then we review the partial results known for hypergraphic sequences in Section 2.3. We introduce new necessary conditions on hypergraphic sequences in Section 3.

2.1. Characterizations of Graphic Sequences. A nonincreasing sequence ϕ = n (d1, d2,...,dn), where i=1 di is even, is graphic if and only if the following equivalent characterizations hold: P

(1) The Erd¨os-Gallai characterization [6] : This is perhaps the most famous and the most cited characterization of graphic sequences.

k n di ≤ k(k − 1) + min{k, di} for all k ∈ [n], (2.1) i=1 X i=Xk+1 which is equivalent to the following system of linear inequalities [21]:

k n di − k(k − 1) ≤ (l − k)k + di for all k,l ∈ [n],l ≥ k. (2.2) i=1 X i=Xk+1 (2) The Ryser characterization [6, 25] : This is one of the oldest characterizations known for graphic sequences. Ryser deﬁnes f = max{i|di ≥ i} and

di +1, if i ∈ [f] d˜i = (di, otherwise.

ϕ = (d1, d2,...,dn) is graphic if and only if (d˜1, d˜2, ..., dñ : d˜1, d˜2, ..., dñ) is bigraphic. We will explain the definition of bigraphic sequences and their characterizations in Section 2.2 under item (2).

(3) The Berge characterization [2] : Berge deﬁnes the sequence d¯= (d¯1, d¯2,..., th d¯n) such that each d¯i is the sum of the i column of the n×n (0, 1)-matrix, th th which has the di leading terms of the i row equal to 1 except for the (i,i) term which is 0. Also the remaining entries are 0. The characterization is

k k di ≤ d¯i for all k ∈ [n]. (2.3) i=1 i=1 X X PROPERTIES OF DEGREE SEQUENCES OF k-UNIFORM HYPERGRAPHS 9

Example 2.1. If d = (3, 2, 2, 2, 1), then the (0, 1)-matrix is equal to 01110 10100 11000 11000   10000   and d¯= (4, 3, 2, 1, 0).  

(4) The Hässelbarth characterization [19, 26] : Define f = max{i|di ≥ i} and th the sequence d´ = (d´1, d´2,..., dń) such that each dí is the sum of the i th column of the n × n (0, 1)-matrix, which has the di leading terms of the i row equal to 1 and the remaining entries are 0. The characterization is k k di ≤ (dí − 1) for all k ∈ [f]. (2.4) i=1 i=1 X X (5) The Bollobás characterization [5] : k n k di ≤ di + min{k − 1, di} for all k ∈ [n]. (2.5) i=1 i=1 X i=Xk+1 X (6) The Grünbaum characterization [15] : k n max{k − 1, di}≤ k(k − 1) + di for all k ∈ [n]. (2.6) i=1 X i=Xk+1 (7) The Fulkerson-Hoffman-McAndrew characterization [11, 13] : k n di ≤ k(n − m − 1) + di (2.7) i=1 i=n−m+1 X X for all k ∈ [n],m ≥ 0, and k + m ≤ n. (8) The Havel-Hakimi Algorithm [18, 17] : Havel and Hakimi, independently introduced an algorithm to recognize graphic sequences, which in known as the Havel-Hakimi algorithm. This algorithm inspired many algorithms for recognition of other classes of graphs, some of which we will see in Section 2.2. Havel-Hakimi algorithm: for n ≥ 1, the nonnegative integer sequence ϕ on size n is graphic if and only if ϕ′ is graphic, where ϕ′ is the list of integers obtained from ϕ by deleting its largest element ∆ and subtracting one from each of the next ∆ largest elements.

(9) The Koren characterization [21] : The Koren characterization is similar to the Fulkerson-Hoﬀman-McAndrew characterization, item (7), only the nonincreasing condition on the sequence has been relaxed.

di − dj ≤ |S|(n − 1 − |T |) (2.8) Xi∈S jX∈T for all S,T such that ∅ 6= S ∪ T ⊂ [n] and S ∩ T = ∅. 10 February 9, 2011

However, it is easy to see that if the inequality 2.8 holds for some S and T , where S consists of vertices with the largest degrees and T consists of vertices with the lowest degrees, then it would hold for any S′ and T ′, where |S′| = |S| and |T ′| = |T |. Also if the inequality does not hold for some S′ and T ′, then it would not hold for S and T , where S consists of the |S′| vertices with the largest degrees and T consists of the |T ′| vertices with the lowest degrees. Therefore, the Koren characterization is equivalent to the Fulkerson-Hoﬀman-McAndrew characterization. Example 2.2. Consider the sequence d = (3, 2, 2, 2, 1) from Example 2.1. Let us check the Koren characterization for S = {v1, v4} and T = {v3, v5}:

di − dj =5 − 3 ≤ |S|(n − 1 − |T |) = 2(5 − 1 − 2)=4, Xi∈S jX∈T which holds. Proposition 3.8, a set of conditions for degree sequences of hypergraphs, in the special case of (simple) graphs is the same as the Koren characterization (see Example 3.11). Therefore, our result can be considered as a generalization of the Koren characterization.

2.2. Degree Sequences of Other Classes of Graphs. In addition to characterizations of degree sequences of (simple) graphs, characterizations of degree sequences of other special classes of graphs are also known. Here we restate some of this information. There have also been some studies on the properties of more complicated classes of graphs that we do not describe here. See [22] for an example.

(1) The Characterization of Degree Sequences of General Graphs: A loop increases the degree of a vertex by two. If we do not restrict our- selves to simple graphs, speciﬁcally allow multiple loops, it is easy to see that: Any nonnegative sequence of integers ϕ = (d1, d2,...,dn) is the degree se- n quence for some graph if and only if i=1 di is even.

(2) Condition on Bigraphic Sequences: P Let P = (p1,...pn) and Q = (q1,...,qm) be sequences of nonnegative integers. The pair (P : Q) is called bigraphic if it is the degree sequence of a bipartite graph. Conditions on sequences to be bigraphic are also known. For example, the Havel-Hakimi algorithm for bigraphic sequences [22, 29] is: The pair of nonnegative sequences of integers (P : Q) is bigraphic if and only if (P ′ : Q′) is bigraphic, where (P ′ : Q′) is obtained from (P : Q) by deleting the largest element pi of P and subtracting one from the pi largest elements of Q.

(3) Condition on Digraphic Sequences: Let P = (p1,...pn) and Q = (q1,...,qn) be sequences of nonnegative integers. The pair (P : Q) is called digraphic if there exists a directed graph G~ for which P is the out-degree sequence and Q is the in-degree sequence. PROPERTIES OF DEGREE SEQUENCES OF k-UNIFORM HYPERGRAPHS 11

Knowing the Havel-Hakimi algorithm for (simple) graphs it is easy to recognize digraphic sequences: The pair of nonnegative sequences of integers (P : Q) is digraphic if and only if (P ′ : Q′) is bigraphic, where (P ′ : Q′) is obtained from (P : Q) by deleting the largest element pi of P and subtracting one from the pi largest elements of Q other than qi.

(4) The Characterization of r-Graphic Sequences: A r-graph is a loopless undirected graph with at most r edges joining a pair of vertices. A sequence is called r-graphic if it is the degree sequence for a r-graph. Based on this deﬁnition (simple) graphs are 1-graphs and graphic sequences could also be called 1-graphic. The characterization of r-graphic sequences has been studied in several articles, including [9]. However, we only state the characterization of a generalization of r-graphic sequences, i.e. f-graphic sequences under item 5.

(5) The Characterization of f-Graphic Sequences: f-graphs are a generalization of r-graphs where the maximum allowed multiplicity of each edge e is defined by f(e). A sequence is called f- graphic if it is the degree sequence for a f-graph. In [13], Fulkerson et. al. introduced a simple characterization, but it holds only for graphs that satisfy a condition on odd-cycles. Tutte [28], presents an answer to the problem of characterization of f-graphic sequences while he is explaining more general results. His conditions are more complicated but they hold for all f-graphic sequences. Here we first give the definitions. Then we will write the Tuttes characterization in a format suitable for our problem. Let G be the graph with all the allowed multiple edges. Let T be any set of vertices of G. Tutte denotes by S(T ) the set of all vertices c of G having the following properties: (a) c is not an element of T . (b) Each edge of G having c as an end has its other end in T . If T does not include every vertex of G denote by k(T ) the number of components H of the induced subgraph G(T ) having the following properties: (a) H has more than one vertex. (b) The sum of the numbers di, taken over all vertices of H, is odd. If T is the set of vertices of G write k(T )=0. Any nonnegative sequence of integers ϕ = (d1, d2,...,dn) is f-graphic if and only if for any subset T of vertices the following inequality holds:

di + k(T ) ≤ di. (2.9) i∈XS(T ) Xi∈T (6) The Characterization of Threshold Sequences: A graph with vertex set {v1, v2,...,vn} is a threshold graph if there exist real weights c1,c2,...,cn n and t such that the 0 − 1 solutions of the inequality i=1 cixi ≤ t are precisely the characteristic vectors of the edges of the graph. The degree sequence of a threshold graph is called a threshold sequenceP . Golumbic [14] and Hammer et. al. [16] showed that: A nonnegative integer sequence d = (d1, d2,...,dn) is a threshold sequence if and only if d = d¯, where the (threshold) sequence d¯ = (d¯1, d¯2,..., d¯n) 12 February 9, 2011

such that each d¯i = n − 1 − di. Threshold graphs have been the subject of several studies. This class of graphs has been used to study the polytope of degree sequences of simple graphs as well as hyper graphs [24, 20]. We discuss hypergraphs and this work in more detail in Section 4.1.2. We should point out here that, the same characterization is valid for the general case of threshold ¯ n−1 k-hypergraphic sequences, where each di = k−1 − di. 2.3. Properties of Hypergraphic Sequences. The characterization of k-hypergraphic sequences is not known [3, 8, 12]. Even for the case k = 3 the problem seems to be diﬃcult [8]. Currently, there is neither a polynomial time algorithm to test 3- hypergraphicness nor a proof that it is a NP-complete problem. Here we gathered some of the known conditions on 3-hypergraphic sequences.

(1) The Choudum Property [9] : Choudum gives a set of sufficient conditions for a sequence to be 3-hypergraphic. This set of conditions is a generalization of the Erdös-Gallai characterization, item (1) in Section 2.1. Let - Mk(ϕ) = max{|E(G)| : G is a k-graph on n vertices, such that degG(vi) ≤ di, for every i, 1 ≤ i ≤ n}, - x+ = max(0, x), and + + + - (x1,...,xn) = (x1 ,...,xn ). Any nonnegative sequence of integers ϕ = (d1, d2,...,dn) is 3-hypergraphic n if i=1 di ≡ 0 (mod 3) , and for all k ∈ [n], k n P k − 1 k − 1 d ≤ k + 2 min d , i 2 j 2 i=1 X j=Xk+1 k k + + M d − ,...,d − . (2.10) k k+1 2 n 2 Choudum has also introduced a formula to calculate Mk(ϕ) in polynomial time. Therefore, one can check this set of conditions in of polynomial time. However, it is easy to find 3-hypergraphs whose degree sequence does not satisfy 2.10.

(2) The Billington Properties [4] : Billington has introduced seven necessary conditions and one suﬃcient condition (algorithm) for sequences to be 3-hypergraphic. Here the notation of the original paper is changed to be consistent with the notation of other references and our paper: (a) Necessary Conditions: If a sequence ϕ = (d1, d2,...,dn) is 3-hypergraphic then the following seven conditions all hold n (i) i=1 di = 3q for some non-negative integer q. Moreover, q is the number of hyperedges in any realization of ϕ. P n−1 (ii) d1 ≤ 2 . (iii) d − d ≤ n−2 . 1 n 2 (iv) For each k ∈ [n], k d =2q + k . i=1 i 3 (v) For each k ∈ [n], k d = q + (n − 1) k − k+1 . Pi=1 i 2 3 P PROPERTIES OF DEGREE SEQUENCES OF k-UNIFORM HYPERGRAPHS 13

(vi) For each k ∈ [n], k k k − 1 n k d ≤ 3 − (d − ) +2 min d , i 3 k 2 j 2 i=1 X h i j=Xk+1 n + min dj , k(k − j − 1) . (2.11) j=Xk+1 (vii) For a hyperedge e = {vi, vj , vk}, let the weight of e be e = i+ j + k. For a sequence ϕ = (d1, d2,...,dn), let the weight of ϕ be n P wt(ϕ)= i=1 idi, and ϕmin(n, q) be the degree sequence of the 3-uniform hypergraph with the q least weighted hyperedges. If a P sequence ϕ = (d1, d2,...,dn) is 3-hypergraphic and 3q = mn+r where 0 ≤ r

Having reviewed the known properties of degree sequences of diﬀerent classes of graphs in this section, we introduce new necessary conditions on hypergraphic sequences, in Section 3. 14 February 9, 2011

3. A Set of Necessary Conditions on Hypergraphic Sequences After a survey on characterization of the degree sequences of diﬀerent classes of graphs in Section 2, in this section, we discuss our result for Problem 1.5, i.e. under n n which conditions a sequence ϕ = (d1, d2, .., dn) ∈ N , where i=1 di ≡ 0 (mod k) is the degree sequence of a k-family? P Our main result in this section (Theorem 3.8) is a set of necessary conditions on a sequence to be k-hypergraphic, as follows: z z dj ≤ αiMi + dj − γiMi /γz+1 αz+1 j∈S i=0 j∈T i=0 X X X X for all S,T such that S ∪ T ⊂ [n] and S ∩ T = ∅.

Mi are parameters measuring the maximum possible number of hyper edges in the class of hyperedges containing αi vertices from S and γi vertices from T . We will explain these parameters in more details in Section 3.1. z is a parameter determining how many of these classes could be used for ϕ. We will explain how to determine z in Section 3.2. We also introduce propositions containing conditions similar or equivalent to this result that we found useful in proving properties of the convex hull polytopes of k-hypergraphic sequences.

Example 3.11 shows how this theorem, for the special case of simple graphs, k=2, is the same as Koren characterization [21], restated below:

di − dj ≤ |S|(n − 1 − |T |), Xi∈S jX∈T for all S,T such that ∅ 6= S ∪ T ⊂ [n] and S ∩ T = ∅. (See Section 2.1 - item 2.8 for the Koren characterization and Example 2.2 for an example of the Koren characterization.)

Therefore, we consider Theorem 3.8 a generalization of the Koren characterization. Although this work had been done independently and the similarity only noticed latter, for the purpose of consistency with the literature we adopt Koren’s notation of S and T sets. Section 4.3 shows our attempt to generalize Koren’s results on graphic sequences and their polytopes to properties of hypergraphic sequences and their polytopes, in Section 3.3.

In this section, the general idea for deciding whether ϕ is k-hypergraphic or not is to check upperbounds on the sum of the degrees of subsets of vertices. The same as in the Koren characterization, for any two distinct subsets of [n], S and T , we obtain an upperbound on the sum of degrees of vertices in S based on the sum of degrees of vertices in T and the number of vertices in S,T and ([n] − S − T ). Each of these upperbounds would be a necessary condition on a (if any) k-family K, for which ϕ is the degree sequence. Therefore they are necessary conditions for ϕ to be k-hypergraphic.

In the rest of this section, ﬁrst, we deﬁne some terms and a (total) order on them in Section 3.1. Then, in Section 3.2, we state and prove our main result on PROPERTIES OF DEGREE SEQUENCES OF k-UNIFORM HYPERGRAPHS 15 the properties of k-hypergraphic sequences, which is a set of necessarily constraints on ϕ to be k-hypergraphic and simplify it for the special case of k = 2. Finally, in

Section 3.3 we state and prove more propositions on the upperbound of i∈S di. P 3.1. An Order on Classes of Hyperedges. As mentioned before, for any two distinct subsets of [n], S and T , we are trying to obtain upperbounds on the sum of degrees of vertices in S, j∈S dj , based on the sum of degrees of vertices in T , d , and the number of vertices in S,T and ([n]−S−T ). The upperbounds are j∈T j P necessarily conditions and are represented in the form of inequalities that should Palways be satisﬁed for a k-hypergraphic sequence and at the same time be as tight as possible.

Hyperedges that contribute the most to the degree of vertices in S and use the least from the degree of vertices in T play a more significant role in determining these upperbounds. To differentiate between hyperedges based on this aspect, we will formally define M(α, γ) and some variations of it to refer to (the number of) hyperedges that contain α vertices from S and γ vertices from T . Then we will define a relation ≺ which is a (total) order on the set of M(α, γ) and each of its variations. In the rest of the text, especially this section, we will use these definitions to state our formulas and proofs. Also for simplicity, we refer to |S|, the cardinality of S, by s and to |T | by t.

Definition 3.1. (M(α, γ):) For fixed n, k, S and T , such that S,T ⊂ [n] and S ∩ T = ∅ , we define ”M(α, γ)”, where 0 ≤ α ≤ s, 0 ≤ γ ≤ t and α + γ ≤ k, to be the number of hyperedges in the complete k-family that contains α vertices from S and γ vertices from T . Note 1: S ∪ T is not necessarly equal to [n].

Note 2: k − α − γ vertices of each hyperedge counted in M(α, γ) are from [n] − (S ∪ T ).

Note 3: M(α, γ) depends on α and γ directly and n, k, S and T indirectly. However, in our work each time we consider the set of M(α, γ) for all α and γ, but ﬁxed n, k, S and T . Therefore, we did not enter n, k, S and T as parameters in the notation of M(α, γ).

Definition 3.2. (relation ≺:) For fixed n, k, S and T , such that S,T ⊂ [n] and S ∩ T = ∅, we define a ”relation ≺”, such that M(α, γ) ≺ M(α′,γ′), if any only if any of the conditions below is held a. α/γ < α′/γ′, b. α/γ = α′/γ′ and α < α′, c. α = α′ =0 and γ>γ′, or d. γ = γ′ =0 and α < α′.

Note 1: When γ = 0, if we consider α/γ to be the positive infinity, equal to any other positive infinity and larger than any other number, case d would fall under 16 February 9, 2011 case b. Therefore, there is no need for d in the definition and cases a through c are sufficient.

Note 2: α = α′ = 0 implies α/γ = α′/γ′.

Note 3: Based on note 1 and 2, the cases for Deﬁnition 3.2 could be written as the following: a’. α/γ < α′/γ′, b’. α/γ = α′/γ′ and α < α′, or c’. α/γ = α′/γ′, α = α′(= 0) and γ>γ′.

Note 4: It is easy to see the relation ≺ as deﬁned in cases a’-c’ satisﬁes the antisymmetry, transitivity, and totality properties on {α/γ|α, γ ∈ N} and therefore it is a total order on {α/γ|α, γ ∈ N}.

Note 5: When α = 0, for any γ, M(α, γ) counts edges that do not contribute to

j∈S dj , and therefore we are not interested in them. However, we included case c in the deﬁnition so that the relation ≺ is total on the set of all M(α, γ). P

Definition 3.3. (ordering on M(α, γ):) For fixed n, k, S and T , such that S,T ⊂ [n] and S∩T = ∅, we define an ordering on the set of M(α, γ),”M0,M1,M2, ..., Ml”, where for all i, j such that i ≤ j, Mj ≺ Mi.

Note: For practicality reasons, the numbering is in the reverse order of a usual numbering on a total set.

Based on the Deﬁnitions 3.1 through 3.3: s n−s−t t 1. M(α, γ)= α k−α−γ γ . 2. The number of hyperedges in the complete k-family is equal to l s n − s − t t n M = = i α k − α − γ γ k i=0 α,γ X X 3. The number of these elements, l +1, is O(k2) and is equal to the number of diﬀerent solutions to the inequality α + γ ≤ k such that 0 ≤ α ≤ s, and 0 ≤ γ ≤ t. 4. The maximum of these elements is M(k, 0), if s ≥ k M0 = M(s, 0), if n − t ≥ k>s M(s, k + t − n), if n − t

5. The minimum of these elements is M(0, k), if t ≥ k Mt = M(0,t), if n − t ≥ k>t M(k + s − n,t), if n − s

 PROPERTIES OF DEGREE SEQUENCES OF k-UNIFORM HYPERGRAPHS 17

To be able to easily refer to α and γ in Mi = M(α, γ), we deﬁne αi and γi as the following:

Definition 3.4. (αi, γi :) For each i, 1 ≤ i ≤ l, when Mi = M(α, γ), we define ”αi” and ”γi” to be, in order, equal to α, and γ . Also we define αt+1 = 0 and γt+1 = ∞.

For each i the number Mi counts the members of a class of hyperedges. Each k- th hyperedge is in the i class, i.e. the class assosiated with Mi, if and only if exactly αi of its verteces are in S and exactly γi of its verteces are in T . Here we define a parameter z to determining how many classes of hyperedges, could be used to obtain j∈S dj . We will later use z in our inequalities for upperbounds on j∈S dj . P P Definition 3.5. (z:) We define z based on the value of j∈T dj as the following:

0, if dj <γ0M0 P j∈T  X r r+1  r, if there exist an r such that 0 ≤ r

Note 2. The third case happens, only if ϕ is not k-hypergraphic. The reverse is not necessarily true.

Note 3. The sum of the degrees of vertices in T for the complete hypergraph is equal to: l l l s n − s − t t s n − s − t t − 1 γ M = γ = t i i i α k − α − γ γ α k − α − γ γ − 1 i=0 i=0 i i i i i=0 i i i i X X X l n − 1 ⇒ γ M = t . i i k − 1 i=0 X This is consistent with the fact that maximum degree for each vertex is equal to n−1 k−1 . Note 4. Deﬁning αt+1 = 0 and γt+1 = ∞ is conveniant for the third case, i.e. z = l.

′ ′′ We need to deﬁne two more parameters, Mi and Mi , to be used in our proof for ′ Theorem 3.8. For a k-family H, Mi counts how many of hyperedges counted in Mi ′′ belong to H. For a sequence ϕ, Mi counts how many of the hyperedges counted in Mi we would use if we were constructing a k-family with maximum possible sum 18 February 9, 2011 of degrees in S, but limited to the sum of the degrees of vertices of T in ϕ . The formal deﬁnition for these parameters is stated in the following:

Definition 3.6. (M ′(α, γ):) For a k-family H on n vertices and sets S and T , such that S,T ⊂ [n] and S ∩ T = ∅, define ”M ′(α, γ)” , where 0 ≤ α ≤ s, 0 ≤ γ ≤ t and α + γ ≤ k to be the number of hyperedges in H containing α vertices in S, and γ vertices in T . Note: H in not necessarily the complete k-family on [n] as it was in Definition 3.1.

We consider the same ordering deﬁned in Deﬁnition 3.3 for M(.)’s on M ′(.)’s.

′′ ′′ Deﬁnition 3.7. (Mi :) We deﬁne Mi based on ϕ, S, and T as the following:

Mi, if i ≤ z z ′′  d − γ M /γ if i = z +1 Mi =  j i i z+1  j∈S i=0  X X 0, if i>z +1   ′ ′′ The α parameter for Mi and Mi is the same as that of Mi, so we use αi to refer to all of them. The same is true for γi.

3.2. Upperbounds and Proofs. After having introduced the deﬁnitions we need in Section 3.1, here we explain and prove our main result on the properties of k- hypergraphic sequences.

Theorem 3.8. For a k-hypergraphic sequence ϕ = (d1, d2, .., dn) and any choice of sets S and T , such that S,T ⊂ [n] and S ∩ T = ∅, the following constrain holds z z dj ≤ αiMi + dj − γiMi /γz+1 αz+1. (3.1) j∈S i=0 j∈T i=0 X X X X To prove Theorem 3.8 we need a lemma.

Lemma 3.9. For any k-hypergraphic sequence ϕ on [n] and any choice of sets S and T , such that S,T ⊂ [n] and S ∩ T = ∅, the following constrain holds: l z+1 ′ ′′ αiMi ≤ αiMi . (3.2) i=0 i=0 X X This lemma might seem obvious based on the informal introduction we gave on ′ ′′ Mi and Mi before deﬁning them. However, it should be proved based on the for- ′ ′′ mal deﬁnitions of Mi and Mi .

Proof. Suppose H is some k-family for which ϕ is the degree sequence. The set of hyperedges of H is a subset of the hyperedges in the complete graph, so ′ ∀i Mi ≤ Mi. (3.3) PROPERTIES OF DEGREE SEQUENCES OF k-UNIFORM HYPERGRAPHS 19

′′ Base on the definition of Mi ′′ ∀i i ≤ z ⇒ Mi = Mi (3.4) Using 3.3 and 3.4: ′ ′′ ∀i i ≤ z ⇒ Mi ≤ Mi (3.5) ′ ′′ ⇒ ∀i i ≤ z ∃mi such that Mi + mi = Mi and mi ≥ 0 ′ ′′ ′ ′′ ⇒ αiMi + αimi = αiMi and γiMi + γimi = γiMi Therefore, z z z ′ ′′ αiMi + αimi = αiMi (3.6) i=0 i=0 i=0 X X X and z z z ′ ′′ γiMi + γimi = γiMi . i=0 i=0 i=0 X X ′ X ′′ On the other hand, by the definitions of Mi and Mi l z+1 ′ ′′ γiMi = ( γidi =) γiMi . i=0 i=0 X Xi∈T X l z ′ ′′ ⇒ γiMi = γz+1Mz+1 + γimi. (3.7) i=z+1 i=0 X X By the definition of the ordering

∀i,j i

Proof. [ Proof of Theorem 3.8 by contradiction] Assume for some k-hypergraphic sequence ϕ = (d1, d2, .., dn) and some sets S and T , such that S ∩ T = ∅ and ∅ 6= S,T ⊂ [n] the inequality (3.1) does not hold. Suppose H is some k-family for which ϕ is the degree sequence. Therefore,

l z z ′ αiMi = dj > αiMi + dj − γiMi /γz+1 αz+1 i=0 j∈S i=0 j∈T i=0 X X X X X l z+1 ′ ′′ ⇒ αiMi > αiMi (3.10) i=0 i=0 X X In the first line of equations (3.10), the equality comes from the definition of ′ Mi and the inequality comes from the assumption of the proof. The r.h.s of the ′′ inequality in the second line is obtained by the definition of Mi . Inequalities (3.2) and (3.10) show a contradiction. Therefore, any k-hypergraphic sequence ϕ should satisfy the inequality (3.1) for any sets S and T , such that S ∩ T = ∅ and S,T ⊂ [n]. Note: There is no need to check Theorem 3.8 for the case S = ∅, because

j∈S dj = 0 and the inequality (3.1) holds.

P See Section A for motivating steps for Theorem 3.8.

Corollary 3.10. If for a sequence ϕ = (d1, d2, .., dn), k ∈ N, and some sets S and T , such that S ∩ T = ∅ and S,T ⊂ [n], the inequality (3.1) does not hold, then ϕ is not k-hypergraphic.

Now, let us simplify the Theorem 3.8 for the special case of k = 2.

Example 3.11. Suppose k = 2. The following table shows the value of Mi, αi, and γi for all i.

i Mi αi γi s 0 M0 = M(2, 0) = 2 2 0 1 M = M(1, 0) = s n−s−t 1 0 1 1 1 2 M = M(1, 1) = s t 1 1 2 11 3 M = M(0, 0) = n−s−t 0 0 i 2 4 M = M(0, 1) = n−s−t t 0 1 i 1 1 5 M = M(0, 2) = t 0 2 i 2 The following table shows the value of z for diﬀerent values of j∈T dj :

i P i γiMi j=0 γjMj condition on j∈T dj z 0 0 0 d < 0 0 P j∈T j P 1 0 0 d < 0 0 Pj∈T j 2 st st 0 ≤ d < st 1 P j∈T j i> 2 – – s.t ≤ d z > 1 P j∈T j P PROPERTIES OF DEGREE SEQUENCES OF k-UNIFORM HYPERGRAPHS 21

Therefore, the conditions in Theorem 3.8 will be as follows:

(1) if z=0: j∈S dj < 0. However this case, i.e. d < 0, does not happen for a sequence of P j∈T j non-negative integers. P

(2) if z=1: j∈S dj ≤ α0M0 + α1M1 + α2 j∈T dj . P s s n P− s − t ⇒ d ≤ 2 +1 + d j 2 1 1 j Xj∈S jX∈T

⇒ dj ≤ s(n − t − 1)+ dj . (3.11) Xj∈S jX∈T (3) if z > 1: j∈S dj ≤ α0M0 + α1M1 + α2M2 +0. P ⇒ dj ≤ s(n − t − 1)+ st = s(n − 1), Xj∈S which is already covered by inequality (3.11) when T = ∅ and therefore t =0

and j∈T dj =0. P Therefore, the only thing that needs to be checked is inequality (3.11).

The system of inequalities presented in (3.11) is the same as the Koren characterization (2.8) [21], which implies that the Koren characterization is a special case of Theorem 3.8 for k = 2.

3.3. More on Upperbounds. In this section, we state and prove more on the upperbounds on i∈S di, most of which are driven from Theorem 3.8 in Section 3.2. We will use these propositions in the next sections to prove some properties and optimize theP running time of algorithms.

Proposition 3.12. If a sequence ϕ is k-hypergraphic, then any permutation ϕ′ of ϕ is also k-hypergraphic.

Proof. Let K be a k-family for which ϕ is the degree sequence. Applying the same permutation on vertices of K results in a k-family K′, for which ϕ′ is the degree sequence.

The system of inequalities (3.1) in Theorem 3.8 contains a lot of redundancies. The following proposition will reduce these redundancies exponentially. Moreover, using Proposition 3.13 results in a better running time of the algorithm in Section ??. 22 February 9, 2011

Proposition 3.13. For any sequence ϕ = (d1, d2, ..., dn) to be k-hypergraphic, the following constraints hold for any s and t such that s,t ≥ 0, s + t ≤ k on ′ ′ ′ ′ ′ ϕ = (d1 ≥ d2 ≥ ... ≥ dn), where ϕ is the non-increasing sorted version of ϕ: s z n z ′ ′ dj ≤ αiMi + dj − γiMi /γz+1 αz+1. (3.12) j=1 i=0 j=n−t+1 i=0 X X X X

Proof. We prove this preposition by proving that it is equivalent to Theorem 3.8. To do so we will show (1) Checking Theorem 3.8 for a sequence ϕ is equivalent to checking the same theorem on the sorted version of the sequence ϕ′. (2) For a sorted sequence ϕ′, the system of inequalities (3.1) in Theorem 3.8 hold if and only if the system of inequalities (3.12) in Proposition 3.13 hold. These two will prove the equivalency of Theorem 3.8 and Proposition 3.13.

Proof of item (1): In Theorem 3.8, since we are checking the constraint, i.e. the system of inequalities (3.1), for all possible subsets of S and T , checking the theorem on a sequence ϕ is equivalent to checking the theorem on any permutation of the sequence, including the sorted version of the sequence in the non-increasing order ϕ′.

Proof of item (2): The system of inequalities (3.12) in Proposition 3.13 is a subset of the system of inequalities (3.1) in Theorem 3.8. Therefore, if the latter hold the former would also hold. On the other hand, for a non-increasing sequence ′ ′ ′ ′ ′ ′ ′ ϕ and ﬁxed s and t, j∈S′ dj is maximum when S = {v1, v2, ..., vs} and j∈T ′ dj in minimum when T ′ = {v′ , v′ , ..., v′ }. Therefore if inequalities (3.1) in P n−t+1 n−t+2 n P Theorem 3.8 hold for any s,t, then inequalities (3.1) in Theorem 3.8 would also hold for any S and T such that |S| = s and |T | = t.

The next proposition is also equivalent to Theorem 3.8 the motivation to seek this format was its similarity to the original Erd¨os-Gallai characterization (2.1), in order to be able to prove some properties for k−graphic sequences similar to thoes Koren proved for the polytope of graphical sequences [21]. See Section 4.3 for properties proved using those preposition.

Proposition 3.14. For a k-hypergraphic sequence ϕ = (d1, d2, .., dn) and any choice of sets S and T , such that S,T ⊂ [n] and S ∩ T = ∅, the following constraints hold l i−1 + dj ≤ min{Mi, ( dj − γj Mj) /γi} αi. (3.13) j∈S i=0 j∈T j=0 X X X X i−1 + Note: If γi = 0, then min{Mi, ( j∈T dj − j=0 γj Mj) /γi} = Mi. Proof. We prove this proposition isP equal toP Theorem 3.8. Consider the possible i−1 + cases for min{Mi, ( j∈T dj − j=0 γj Mj ) /γi}, we show that the same amount P P PROPERTIES OF DEGREE SEQUENCES OF k-UNIFORM HYPERGRAPHS 23 would be added to the upperbound on j∈S dj in both Proposition 3.14 and The- orem 3.8: i−1 P + (1) If min{Mi, ( dj − γj Mj) /γi} = Mi j=0 jX∈T X i−1 ⇐⇒ 0 ≤ Mi ≤ ( dj − γj Mj )/γi j=0 jX∈T X i−1 ⇐⇒ γiMi ≤ dj − γj Mj j=0 jX∈T X i ⇐⇒ γj Mj ≤ dj j=0 X jX∈T ⇐⇒ i ≤ z.

Therefore, for this i, as in Theorem 3.8, Mi will be added to the upperbound for j∈S dj . i−1 i−1 P + (2) If min{Mi, ( dj − γj Mj) /γi} = ( dj − γj Mj )/γi (6= Mi) j=0 j=0 jX∈T X jX∈T X i−1 ⇐⇒ 0 ≤ ( dj − γjMj )/γi

n z Therefore, for this i, as in Theorem 3.8, j=n−t+1 dj − i=0 γiMi /γz+1 αz+1 will be added to the upperbound for j∈PS dj . P i−1 i−1 + P (3) If min{Mi, ( dj − γj Mj) /γi} = 0 (6= ( dj − γj Mj )/γi) j=0 j=0 jX∈T X jX∈T X i−1 ⇐⇒ ( dj − γj Mj )/γi < 0 ≤ Mi j=0 jX∈T X i−1 ⇐⇒ dj < γj Mj j=0 jX∈T X ⇐⇒ zz +1. Therefore, for this i, as in Theorem 3.8, 0 will be added to the upperbound

for j∈S dj . P 24 February 9, 2011

4. Polytope of Degree Sequences As we in saw Section 2.1, most of the well-known characterizations of graphic sequences are systems of (redundant) inequalities. Each system could be used to determine a polytope in Rn. These polytopes and their properties, such as volume, extreme points and adjacency of points, have been the subject of several studies. The concept of polytopes has been also extended for k-hypergraphic sequence. A question that rises naturally is wheather there are any holes in this polytope for k-hypergraphic sequences. (See Section 1.3 for basic deﬁnitions.)

In this section, we investigate the properties of Dn(k), especially the Problem 1.12, i.e. whether there are any holes in the convex hull of the k hypergraphic sequences, Dn(k)? For the case k = 2, Koren showed that Dn is the polytope determined by the systems of inequalities (2.8) of the Koren characterization [21] and therefore, there is no hole in Dn. Forthere more, Dn could be determined by each one of the systems of inequalities in Section 2.1 (For examples of the proof see Theorem 4.2 in [24] or theorem 1.1 in [23].) However, for k > 2 it is not known whether there exists a hole in Dn(k) or not. There has been some speculation that for k> 2 there are non-degree sequence holes within the convex hull Dn(k) of degree sequences. However, we know of no such example [20].

First we survey the known properties of Dn(2) and the general case of Dn(k) in order in Sections 4.1 and 4.2. Then in Section 4.3, we introduce and prove new properties of Dn(k), generalizing known properties of Dn to Dn(k). Finally, in Section ?? we explain some statistical results gained by computer programming and our results from Section 3.

4.1. Properties for Dn. In this section we survey the known properties of Dn(2).

4.1.1. Koren Properties for Dn. Koren showed a large number of properties of Dn [21]. Here we restate these properties in three categories. In Section 4.3 we will state and prove the generalization of some of these properties for Dn(k). In order to be consistant with the rest of our text, we have made slit changes in Koren’s notation. The main one is that we use Dn instead of Kn.

A. Elementary Properties of Dn:

(1) (0, ..., 0) ∈ Dn.

(2) If (a1, ..., am) ∈ Dn, then 0 ≤ ai ≤ n − 1 (i = l, ..., n). (Hence Dn, is a bounded polyhedral set, i.e., a convex polytope.)

′ ′ ′ (3) Let ϕ = (al, ..., an) ∈ Dn . If ϕ = (a1, ..., an) is a permutation of ϕ then ′ ϕ ∈ Dn.

(4) Dn, is centrally symmetric with respect to the point n − 1 n − 1 ,..., 2 2 PROPERTIES OF DEGREE SEQUENCES OF k-UNIFORM HYPERGRAPHS 25

i.e., if (a1, ..., an) ∈ Dn, then (n − 1 − a1, ..., n − 1 − an) ∈ Dn.

(5) D1 is 0-dimensional. D2 is 1-dimensional. For n> 2, Dn, is n-dimensional.

B. Properties of the vertices of Dn:

(1) If ϕ = (a, ..., a) ∈ Dn, then ϕ is a vertex of Dn, iﬀ a =0 or a = n − 1.

(2) Let ϕ = (a1, ..., an) be a non-increasing realizable sequence. For every i, 1

{Pk :1 ≤ k ≤ ai}, if ai < i,

{Pk :1 ≤ k ≤ ai +1, k 6= i}, if ai ≥ i.

Remark: item 2 is identical to Theorem 3.2 in [11].

(3) Let (a1,...,an) ∈ K, and suppose that n − 1 >a1 ≥ a2 ≥ . . . ≥ an. If, for a certain k, there is equality in condition (2.1), then an < k.

(4) Let (a1, ..., an) ∈ K, and suppose that n − 1 >a1 ≥ a2 ≥ . . . ≥ an > 0, and a1 > an. Let h be the largest index i for which a1 = ai, (1 ≤ h

(5) Let ϕ = (a1,...,an) ∈ D(n). If n − 1 >a1 ≥ a2 ≥ . . . ≥ an > 0, then ϕ is not a vertex of D(n).

(6) (Theorem 1 in [21]:) A sequence ϕ = (a1,...,an) of real numbers is a vertex of D(n) if and only if ϕ has a unique realization.

(7) If ϕ = (a1,...,an) is a vertex of D(n), then (a) Each ai (i = l, ..., n) is an integer. n (b) i=1 ai is even (and therefore ϕ is realizable). (c) The realization of ϕ is unique. P (8) Let ϕ = (a1,...,an) be a realizable sequence. If ϕ is not a vertex of D(n), then ϕ has more than one realization.

C. Characterization of Vertices of Dn:

(1) Let

n1 n2 nm

ϕ = (b1, ..., b1, b2, ..., b2, ..., bm, ..., bm), m b1 >b2 >...>bm >z 0,}| {i=1z n1}|= {n, niz> 0}| for{ 1 ≤ i ≤ m. Suppose ϕ is a vertex of D(n). Then b1 = n − 1, and if m > 1 then bm = n1. If m> 2 P then b2 = n − 1 − nm. 26 February 9, 2011

(2) Let n1 nm

ϕ = (b1, ..., b1, ..., bm, ..., bm), m b1 > b2 >...>bm > 0,z }|i=1{ n1 =z n}|, m{ > 3, ni > 0 for 1 ≤ i ≤ m, b1 = n − 1, b2 = n − 1 − n1, bm = n1. Deﬁne P n1+n2 n3 nm−1 ∗ ϕ = (b2, ..., b2, b3, ..., b3 ..., bm−1, ..., bm−1), ∗ Under these conditionsz }| ϕ{ zis a}| vertex{ z of D(}|n − nm{) if and only if ϕ is a vertex of D(n).

(3) (Theorem 2 in [21]:) Let

n1 nm

ϕ = (b1, ..., b1, ..., bm, ..., bm), m b1 > b2 >...>bm > 0, z i}|=1 n{1 = nz , n}|i > 0{ for 1 ≤ i ≤ m. Under these conditions ϕ is a vertex of D(n) if and only if: P b1 = n − 1 Bm = n1 b2 = n − 1 − nm Bm−1 = n1 + n2 . . . . bl = n − 1 − nm − . . . − nm−l+2 Bm−k = n1 + n2 + . . . + nk+1, m+1 m−2 where l = [ 2 ], k = [ 2 ]. (Hence m − k = l1.)

Note that bl > bl+1(l +1 = m − k) if and only if nk+2 ≥ 2 (k +2 = m − l − 1 = [m/2] + 1).

(4) (Theorem 3 in [21]:) There are exactly 2n−1 non- increasing sequences (a1, ..., an) which are vertices of D(n).

4.1.2. The polytope of degree sequences. Peled and Srinivasan studied the polytope of degree sequences of simple graphs [24]. Using linear programming duality and the structure of threshold graphs, Peled and Srinivasan reproduced some of the properties of D(n) introduced by Koren in [21], listed in Section 4.1.1 above. They also introduced several other properties of D(n), we review some of the most important ones in this section. However, we shall ﬁrst deﬁne some of the terms they used, i.e. threshold sequence and majorization.

Threshold graphs have been the subject of several studies. These class of graphs have been used to study the polytope of degree sequences of simple graphs as well as hypergraphs (see in order [24] and [23] for examples.)

Deﬁnition 4.1. (threshold graph:) A graph G = ([n], E) is called a ”threshold graph” if there exists an n-tuple c = (c1,c2,...,cn) of real numbers such that for n all X ∈ 2 , we have X ∈ E if and only if i∈X ci ≥ 0.

The cis are called weights and we sayP that G is determined by the weights (c1,c2,...,cn). PROPERTIES OF DEGREE SEQUENCES OF k-UNIFORM HYPERGRAPHS 27

Deﬁnition 4.2. (threshold sequence:) The degree sequence of a threshold graph is called a ”threshold sequence”.

See Section 2.2 item 6 for the characterization of threshold sequences. See Defini- tion 4.5 and Definition 4.6 for the generalization of these definitions for hypergraphs.

Deﬁnition 4.3. (majorization:) For any real sequence f = (f1,...,fn), let f[1] ≥ f[2] ≥ . . . ≥ f[n] denote the components of f sorted in a nonincreasing order. We k k say that f ”majorizes” g, denoted f g, when i=1 f[i] ≥ i=1 g[i] holds for each k =1,...,n, with equality holding for k = n. P P

Note: majorization is also known as dominance order. Deﬁnition 4.4. (strict majorization:) We say that f ”strictly majorizes” g, denoted f ≻ g, when f g and at least one of the above inequalities for k =1,...,n−1 is strict. Note: f strictly majorizes g means that the common sum of the components of f and g is distributed more evenly in g than in f.

Now we are ready to recall important properties of Dn, introduced by Peled and Srinivasan.

(1) (Theorem 2.8 in [24]:) A degree sequence is an extreme point of Dn, if and only if it is a threshold sequence.

(2) (Theorem 3.5 in [24]:) Let T1 = (V, E1) and T2 = (V, E2) be the (unique) realizations of the threshold sequences f and g, respectively. Then f and g are adjacent extreme points of Dn, if and only if |E1 ⊕ E2| = 1. |E1 ⊕ E2| = 1 means that the two sets E1 and E2 have only one dif- ference, i.e. the only diﬀerence between them is that one contains an edge that the other one does not.

(3) (Theorem 3.7 in [24]:) Let (f1,...,fn) and (g1,...,gn) be threshold sequences. Then they are adjacent extreme points of Dn, if ad only if there th exist indices i 6= j such that f − g = ±(ui + uj). (ut is the t unit vector.)

(4) Let f = (f1,...,fn) be any degree sequence. Let δ1 < δ2 <... <δm, be the distinct nonzero integers occurring in f. Set δ0 = 0. The degree partition of f is deﬁned as D = (D(0),...,D(m)), where D(i) = {j|fi = δj }, 0 ≤ i ≤ m. Also deﬁne di = |D(i)|. (Theorem 3.8 in [24]:) Let f = (f1,...,fn) be a threshold sequence. Then ′ ′ ′ the extreme points f = (f1,...,fn) of Dn, adjacent to f are given by ′ (a) f = f + ui + uj, where m − 1 i ∈ D(r), j ∈ D(m − r), and 0 ≤ r ≤ 2 or j k m +1 i, j ∈ D , i 6= j, m even; 2 j k 28 February 9, 2011

′ (b) f = f − ui − uj, where m i ∈ D(r), j ∈ D(m − r + 1), and 0 ≤ r ≤ 2 or j k m +1 i, j ∈ D , i 6= j, m odd. 2 j k

(5) (Theorem 3.9 in [24]:) Let f = (f1,...,fn) be a threshold sequence with a degree partition D = (D(0),...,D(m)). Then the number of extreme points of Dn, adjacent to f is given by − ⌊ m 1 ⌋ ⌊ m ⌋ 2 2 d d d + d d + q , (4.1) i m−i i m−i+1 2 i=1 i=1 X X m +1 where q = . 2 (6) (Theorem 4.2 in [24]:) Dn is determinedj k by the linear inequalities

di − dj ≤ |S|(n − 1 − |T |) Xi∈S jX∈T for all S,T such that ∅ 6= S ∪ T ⊂ [n] and S ∩ T = ∅.

(7) (Theorem 4.4 in [24]:) For n ≥ 3, the facets of Dn, are given by: (a) di ≥ 0, i =1,...,n (only if n ≥ 4). (b) di ≤ n − 1, i =1,...,n (only if n ≥ 4). (c) i∈S di − i∈T di ≤ |S|(n−1−|T |) for all sets S,T such that S,T 6= ∅, S ∪ T ⊂ [n] and |S ∪ T | =2, 3,...,n − 3,n. P P (8) (Theorem 5.8 in [24]:) A degree sequence ϕ is threshold if and only if ϕ is not strictly majorized by any degree sequence.

(9) (Corollary 5.9 in [24]:) Every degree sequence is a convex combination of the rearrangements of a single threshold sequence.

4.2. Polytope of Hypergraphic Sequences. Murthy and Srinivasan [23] studied the polytope of degree sequences of hypergraphs. Their main results concern the extreme points and facets of Dn(k). In this section we summarize their results in three categories..

A. Extreme Points of Dn(k):

Given an n-tuple c = (c1,c2,...,cn) of real numbers and X ⊆ [n] we denote:

c(X)= ci. iX∈X Deﬁnition 4.5. (k-threshold hypergraph:) A k-hypergraph H = ([n], E) is called a ”k-threshold hypergraph” if there exists an n-tuple c = (c1,c2,...,cn) of real n numbers such that for all X ∈ k , we have X ∈ E if and only if c(X)= i∈X ci > 0. P PROPERTIES OF DEGREE SEQUENCES OF k-UNIFORM HYPERGRAPHS 29

Definition 4.6. (k-threshold sequence:) The degree sequence of a k-threshold graph is called a ”k-threshold sequence”. Definition 4.5 and Definition 4.6 are generalizations of Definition 4.1 and Defi- nition 4.2.

Deﬁnition 4.7. (

Deﬁnition 4.8. (q-ideal:) Let

(1) (Theorem 2.5 in [23]:) A degree sequence is an extreme point of Dn(k) if and only if it is a k-threshold sequence. (This is a generalization of item (1) in Section 4.1.2.)

See Section 3 of [20] for a survey on properties of threshold hypergraphs and therefore the extreme points of Dn(K).

(2) (Corollary 2.6 in [23]:) k-threshold sequences are uniquely realizable.

(3) (Theorem 2.8 in [23]:) Let T1 = (V, E1) and T2 = (V, E2) be the (unique) realizations of the k-threshold sequences f and g, respectively. Then f and g are adjacent extreme points of Dn(k), if and only if |E1 ⊕ E2| = 1. (This is a generalization of item (2) in Section 4.1.2.)

(4) (Theorem 2.9 in [23]:) The diameter of Dn(k), i.e. the maximum distance n of two (extreme) points of Dn(k), is k . By distance from point a to point b we mean the minimum number of traverses from a point to one of its adjacent points needed to reach b from a. In particular, the distance between adjacent points is 1.

(5) (Theorem 2.10 in [23]:) Let T1 = (V, E1) and T2 = (V, E2) be the (unique) realizations of the k-threshold sequences f and g, respectively. Then the distance between f and g in Dn(2) is |E1 ⊕ E2|. Note: This Theorem is stated only the for special case of k = 2.

(6) (Theorem 2.12 in [23]:) Let H = ([n], E) be a K-threshold hypergraph. Assume that H is determined by the weights (c1,...,cn) and let φ be a 30 February 9, 2011

permutation of [n] satisfying c(1) ≥ c(2) ≥ . . . ≥ c(n) . Deﬁne a linear order

B. Facets of Dn(k):

Deﬁnition 4.9. (Fk(a):) Let n, k be positive integers with n ≥ k. Given an n-tuple a = (a1,...,an) of integers, deﬁne

Fk(a)= a(X), XX [n] where the sum is over all X ∈ k satisfying a(X) > 0. Deﬁnition 4.10. (Mk(a):)Let n, k be positive integers with n ≥ k. Let a = (a1,...,an) be an integral n-tuple. Deﬁne ”Mk(a)” to be the n column matrix whose rows are the characteristic vectors of k-subsets X of [n] satisfying a(X)=0.

(1) (Theorem 3.1 in [23]:) Let n, k be positive integers with n ≥ k. Then the polytope Dn(k) is n dimensional.

(2) (Theorem 3.8 in [23]:) Let a = (a1,...,an) be an integral n-tuple and let b be a real number. Let n ≥ k + 1. The linear inequality

a1x1 + a2x2 + . . . + anxn ≥ b

is a facet of Dn(k) if and only if b = Fk(a) and rank(Mk(a)) = n − 1.

(3) (Theorem 3.10 in [23]:) Let n ≥ k+1 and let a = (a1,...,an) be an integral n-tuple with g.c.d(a1,...,an) = 1 and such that the linear inequality

a1x1 + a2x2 + . . . + anxn ≥ Fk(a) n/2 determines a facet of Dn(k). Then |ai|≤ nr , i =1,...,n and n F (a) ≥ n r(n+2)/2. k r (4) (Theorem 3.11 in [23]:) For n ≥ 3, the facets of Dn(2), are given by:

di − di ≤ |S|(n − 1 − |T |) Xi∈S Xi∈T for all sets S,T ⊆ [n] such that S ∩ T = ∅ satisfying one of the following conditions: (a) |S ∪ T | = 1, |S ∪ T |≥ 3. (b) |S|≥ 1, |T |≥ 1, |S ∪ T |≥ 0 or |S ∪ T |≥ 3. PROPERTIES OF DEGREE SEQUENCES OF k-UNIFORM HYPERGRAPHS 31

Note: This theorem is stated for the special case of k = 2. It is similar to Theorem 4.4 in [24], item 7 in Section 4.1.2.

(5) (Theorem 3.12 in [23]:) Let n,p,q,k be positive integers with n ≥ k + 1. Let S,T ⊆ [n] with S ∩ T = ∅. The inequality

pdi − qdi ≤ Fk(p,...,p, −q,..., −q, 0,..., 0) Xi∈S Xi∈T deﬁnes a facet of Dn(2) whenever any of the following conditions holds: (a) p = q = 1, |S ∪ T | = 1, |S ∪ T |≥ k + 1. (b) p + q = r, |S ∪ T | = 0, p = 1, |T | = 1, |S| = q + 1. (c) p + q = r, |S ∪ T | = 0, q = 1, |S| = 1, |T | = p + 1. (d) p + q = r, |S ∪ T | = 0, |T | = p + 1, |S| = q + 1. (e) p + q ≤ r, |S ∪ T |≥ k + 1, p = q = 1, |S| = |T | = 1. (f) p + q ≤ r, |S ∪ T |≥ k + 1, p = 1, |T | = 1, |S| = q + 1. (g) p + q ≤ r, |S ∪ T |≥ k + 1, q = 1, |S| = 1, |T | = p + 1. (h) p + q ≤ r, |S ∪ T |≥ k + 1, |T | = p + 1, |S| = q + 1.

(6) (Theorem 3.14 in [23]:) For n ≥ 1, the inequality n n fixi − fiyi ≤ F3(f1,f2,...,fn,f1,f2,...,fn, 0, 0, 0) i=1 i=1 X X is facet inducing for D2n+3(3). (f0 = 0, f1 = 1 and fn = fn1 + fn2 are Fibonacci numbers.) Note: This theorem is stated for the special case of k = 3.

The original reference [23] also contains some properties of incidence matrices and calculations of facets of D4(3),D5(3), and D6(3). However, since those properties are less relevance to the topic of this paper, we do not recall them here.

4.3. Generalizing Koren Properties for Dn(k). In this section we generalize some properties of Dn introduced by Koren to Dn(k), using the necessary condition and propositions introduced and proved in Section 3.

A. Elementary Properties of Dn(k):

(1) (0, ..., 0) ∈ Dn(k).

Proof. (0, ..., 0) ∈ Dn(k), because the empty k-family on [n] is the realization of (0, ..., 0).

n−1 (2) If (a1, ..., an) ∈ Dn(k), then 0 ≤ ai ≤ k−1 (i = l, ..., n).

Proof. Let T = i,S = ∅; then from Theorem 3.8 we obtain di ≤ 1.M(1, 0) = n−1 k−1 . Let S = i,T = ∅; then from Theorem 3.8 we obtain −di ≤ 0.M(0, 1) = 0. n−1 n−1 The complete k-family is the realization of ( k−1 ,..., k−1 ) and (0, ..., 0) is the realization of (0, ..., 0), therefore the bonds on a are tight. i 32 February 9, 2011

′ ′ ′ (3) Let ϕ = (al, ..., an) ∈ Dn(k). If ϕ = (a1, ..., an) is a permutation of ϕ then ′ ϕ ∈ Dn(k). Proof. Let H be a k-family for which ϕ is the degree sequence. Applying the same permutation on vertices of H results in a k-family H′, for which ϕ′ is the degree sequence.

(4) Dn(k), is centrally symmetric with respect to the point n−1 n−1 k−1 ,..., k−1 2 2 n−1 n−1 i.e., if (a1, ..., an) ∈ Dn(k), then ( k−1 − a1, ..., k−1 − an) ∈ Dn(k). n−1 n−1 Proof. ( k−1 −a1, ..., k−1 −an) is the degree sequence of the complement of any k-family that realizes (a , ..., a ). 1 n n−1 n−1 (5) (corollary:) The number of realizations of (a1, ..., an) and ( k−1 −a1, ..., k−1 − a ) as k-families are equal. n

(6) For n < k, Dn(k), is 0-dimensional. For n = k, Dn(k), is 1-dimensional. For n > k, Dn(k), is n-dimensional.

n−1 Proof. For n < k, k−1 = 0. Using property (A.2), for each (a1, ..., an) ∈ D (k), 0 ≤ a ≤ 0 for all 1 ≤ i ≤ n. Therefore, D (k)= {(0,..., 0)} which n i n represents the empty k-hypergraph on [n]. n−2 For n = k, consider S = {vj} and T = {vi}, dj ≤ k−2 di = di. Therefore for any i, j ∈ [n], d = d . On the other hand, n−1 = 1 and using prop- i j k−1 erty (A.2), for each (a1, ..., an) ∈ Dn(k), 0 ≤ ai ≤ 1. Therefore the only n−1 n−1 k-hypergraphic sequences are (0,..., 0) and ( k−1 ,..., k−1 )=(1,..., 1) and Dn(k) is the line segment between these two points. n−1 If n > k, for1 ≤ i ≤ n the set of all (a1, ..., an), were ai = k−1 and the rest of coordinates are n−2 are a set of independent vectors. Each of these vec- k−2 tors represents the k-hypergraphs with all possible hyperedges containing vi and no hyperedge without vi. Therefore Dn(k), is n-dimensional.

B. Properties of the vertices of Dn(k):

n−1 (1) If ϕ = (a, ..., a) ∈ Dn(k), then ϕ is a vertex of Dn(k), iﬀ a =0or a = k−1 . Proof. This follows easily from properties (A.2) and (A.4), because (a, ..., a) n−1 n−1 on the line segment between (0,..., 0) and ( k−1 ,..., k−1 ), which both are vertices of D (k). n (2) Let ϕ = (a1, ..., an) be a non-increasing realizable sequence. For every i, 1

{v1,...,vai }, if ai < i, Vi = ({v1,...,vi−1, vi+1,...vai+1}, if ai ≥ i.

For every i,1

[n]/vi Proof. Ui contains the ﬁrst ai members of the family of sets k−1 in lex- icographical order. Sort members of U in lexical order. Let H = ([n], E) i be a realization of ϕ, and u be the ﬁrst member of Ui in lexical order for which {vi ∪ u} is not a k-hyperedge of H. Then there exist a k-hyperedge ′ ′ ′ ′ u such that vi ∈ u and u ∈ E, but {u /vi} ∈/ Ui

′ Consider the smallest j such that vj ∈ u, vj ∈/ u . Since in the lexico- ′ graphical order u is before u /vi, there exist a vertices vk such that j < k, ′ vk ∈ u , and vk ∈/ u. Since j < k, in ϕ and therefore in H, dj ≥ dk, so these ′ exist a k-hyperedge e in E such that vj ∈ e, vk ∈/ e. Delete vk from u ; add vj to it. Delete vj from e; add vk to it. Repeat this process for the next j, ′ until u has converted to u ∪ vi. Then move on to the next u, until there is no such u. n−1 (3) Let ϕ = (a1,...,an) ∈ Dn(k), and suppose that k−1 > a1 ≥ a2 ≥ . . . ≥ a > 0. If, for a certain S and T 6= ∅, there is equality in the condition of n Theorem 3.8, then

sγz+1 n − 1 n − t − 1 dj < − . αz+1 k − 1 k − 1 jX∈T h i Moreover,

sγz+1 n − 1 n − t − 1 dn < − . tαz+1 k − 1 k − 1 h i Proof by contradition. : Consider d ≥ sγz+1 n−1 − n−t−1 . j∈T j αz+1 k−1 k−1 n−1 n−1 We know Pk−1 >a1 ⇒ s hk−1 > j∈S dj .i Equality in Theorem 3.8 means: P z z dj = αiMi + dj − γiMi /γz+1 αz+1. j∈S i=0 j∈T i=0 X X X X Consider x be the largest index for which γx = 0. Based on the deﬁnitions of Mx, γy = 0 if and only if y ≤ x. Since j∈T dj > 0, we have x ≤ z. Each vertex in S is in n−t−1 hyperedges not containing any vertex of T . k−1 P Therefore edges with no vertex in T contribute s n−t−1 to d and k−1 j∈S j z α M and zero to d and therefore zero to z γ M . There- i=0 i i j∈T j i=0 Pi i fore z α M = n−t−1 + z α M and z γ M = z γ M . P i=0 i i k−P1 i=x i i i=0 iP i i=x i i Putting this all together we have: P P P P z z n − 1 s > d = α M + d − γ M /γ α k − 1 j i i j i i z+1 z+1 i=0 i=0 Xj∈S X jX∈T X z z n − t − 1 = + α M + d − γ M /γ α k − 1 i i j i i z+1 z+1 i=x i=x X jX∈T X z z n − t − 1 α α = + α M + z+1 d − z+1 γ M k − 1 i i γ j γ i i i=x z+1 z+1 i=x X jX∈T X 34 February 9, 2011

For all i, j such that i ≤ j we have αi ≥ αj . Therefore, αz+1 z γ M ≤ γi γj γz+1 i=x i i z αi z ( γiMi)= αiMi. i=x γi i=x P P n − 1 n −Pt − 1 z α z ⇒ s > + α M + z+1 d − α M k − 1 k − 1 i i γ j i i i=x z+1 i=x X jX∈T X n − t − 1 αz+1 =s + dj k − 1 γz+1 jX∈T n − t − 1 α sγ n − 1 n − t − 1 ≥s + z+1 z+1 − k − 1 γz+1 αz+1 k − 1 k − 1 h i n − t − 1 n − 1 n − t − 1 ≥s + s − s k − 1 k − 1 k − 1 n − 1 =s k − 1 n − 1 n − 1 ⇒ s >s . k − 1 k − 1 Which is a contradiction. Therefore, d < sγz+1 n−1 − n−t−1 j∈T j αz+1 k−1 k−1 and since d is the smallest in the degree sequence then dh< sγz+1 n−1 −i n P n tαz+1 k−1 n−t−1 h k−1 . i Note: Without separating edges that do not contribute to j∈T dj we have: P sγz+1 n − 1 sγz+1 n − 1 dj < and dn < . αz+1 k − 1 tαz+1 k − 1 jX∈T PROPERTIES OF DEGREE SEQUENCES OF k-UNIFORM HYPERGRAPHS 35

5. Reduction to the Problem of Lattice Points In this section, we deﬁne a set of relations ψ. Using these relations, we reduce the problem concerning holes in the polytope of degree sequences to a problem on lattice points. We also use these relations to introduce some interesting properties of the extreme points of the polytopes of hypergraphic sequences.

Let us ﬁrst recall some deﬁnitions from Section 1.3.

(1) Fn(k) is the set of all f-hypergraphs on the set of [n] (Definition 1.6.) (2) DSn(k) is the set of degree sequences of members of Fn(k) (Definition 1.7.) (3) Dn(k) is the convex hull polytope of DSn(k) (Definition 1.8.) (4) A point d = (d1, d2, ..., dn) ∈ Dn(k) is an “integer point” of Dn(k) if (a)for n all i ∈ [n], di ∈ N, and (b) i=1 di ≡ 0 (mod k) (Definition 1.9.) P In Section 5.1 we define some of the preliminary operations and concepts we need. In Section 5.2, we introduce a bijection ψF from Fn(k) to Fn−1(k−1)×Fn−1(k) and reduce the problem concerning holes in the polytope of degree sequences (Problem 1.12) to a problem on lattice points (Problem 5.9). Finally, in Section 5.3 introduce some interesting properties of threshold graphs and therefore, those of the extreme points of the polytopes of hypergraphic sequences.

5.1. The Lift Operation and Hidden Points. In this subsection, we deﬁne some of the preliminary operations, state some of their properties and deﬁne some concepts. We will use these material in the rest of this Section 5.

Deﬁnition 5.1. (zero-pad operation Z:) For any nonnegative integers n deﬁne the “zero-pad operation Z” on any point p = (p1,p2,...,pn) to add another dimension to p and set the value of the n +1 dimension to 0 , i.e. Z : Rn → Rn+1 Z(p)= p ; 0 .

We call Z(p) = (p1,p2,...,pn, 0) “zero-padded image” of p.

It is easy to generalize the concept of the zero-pad operation to higher dimensional objects. For each P ⊂ Rn let Z(P ) be the set of the zero-padded images of all points of P , i.e. Z(P )= {Z(p)|p ∈ P }.

Similarly, we define a lift operation. Definition 5.2. (lift operation Lk:) For any nonnegative integers k,n define the k “lift operation L ” on any point p = (p1,p2,...,pn) to add another dimension to p n and set the value of the n +1 dimension to i=1 pi/k , i.e. k n n+1 L : R → R P n k L (p)= p ; pi/k . i=1 X 36 February 9, 2011

1 1 1 0.4 0.8

0.6 0.3 0.5 0.5 0.4 0.2

0 0.2 0 0.1 0 0 −0.5 −0.5 −0.2 −0.1

−0.4 −0.2 −1 −1 1 −0.6 1 −0.3 −0.8 0 0 −0.4 1 1 −1 0 −1 0 −1 −1 (b) −1 (a)

Figure 2. (a) The zero-padded image of the unit cycle. (b) the result of L3 on the unit cycle.

n We call L(p) = (p1,p2,...,pn, i=1 pi/k) “k-lifted image” of p.

It is also easy to generalize the conceptP of the lift operation to higher dimensional objects. For each P ⊂ Rn let Lk(P ) be the set of the lifted images of all points of P , i.e. Lk(P )= {L(p)|p ∈ P }.

See Figure 2.(a) for the zero-padded 2-dimensional unit cycle and Figure 2.(b) for the result of L3 on this unit cycle.

We can apply the zero-pad and lift operations on Dn(k) and DSn(k). For simplic- k k ity, let us donate Z(Dn(k)), Z(DSn(k)), L (Dn(k)) and L (DSn(k)) respectively by ZDn(k), ZDSn(k), LDn(k) and LDSn(k).

Now consider the lifted points and objects in Rn+1 on a grid (of (n + 1)- dimensional unit hypercube with respect to the origin.) It is easy to prove the following properties for the zero-pad and lift operations:

(1) Z and Lk are not surjective. (2) For each point p = (p1,p2,...,pn), Z(p) is a grid point if and only if p is a grid point. k (3) For each point p = (p1,p2,...,pn), L (p) is a grid point if and only if p is n a grid point and i=1 pi = 0 (mod k). (4) Zero-padded and lifted images of any object P in Rn, are objects with the same number of dimensionsP in Rn+1. (5) Lk(P ) is Z(P ) in the (n + 1)-dimensional space, rotated around the hyper- n plain i=1 xi = 0 with an angle equal to Arctan(1/k), where xi refers to the i-coordinate in the space. (see Figure 2.) P PROPERTIES OF DEGREE SEQUENCES OF k-UNIFORM HYPERGRAPHS 37

k (6) The L operation rotates Dn(k) so that the integer points are mapped to grid points. (7) For any n, k if there is no hole in Dn(k) , then there is no hole in ZDn(k) and LDn(k). (8) zero-padding Dn−1(k) by Z can be translated by adding a vertex vn (with no hyperedges) to the vertex set of all (k)-families in DSn−1(k), which will be used in ψ relations introduced in Section 5.2. k−1 (9) Lifting Dn−1(k − 1) by L can be translated by adding a vertex vn to the vertex set of all (k − 1)-families in DSn−1(k − 1) and adding vn to all of the hyperedges of these hypergraphs, which will be used in ψ relations introduced in Section 5.2. b (n,k) (n,k) (n,k) (10) For each point p ∈ Dn(k) if p = i=1 αixi for some points x1 , x2 , (n,k) ..., xb and some positive real numbers α1, α2,...,αb, then Z(p) = b (n,k) P b (n,k) i=1 αiZ xi ∈ ZDn(k) and L(p)= i=1 αiL xi ∈ LDn(k). P P 5.2. A Bijection and Reduction to Lattice points. In this section we introduce a bijection ψF : Fn(k) → Fn−1(k − 1) × Fn−1(k) and prove that ψF is a bijection. We also introduce relations ψ and ψ similar to ψ . Finally, we DS D F reduce the problem concerning holes in the polytope of degree sequences (Problem 1.12) to a problem on lattice points (Problem 5.9), using these relations.

In order to introduce the bijection ψF , we show that each k-family on [n] can be mapped into a pair of hypergraphs, a k-family and a (k − 1)-family both on [n − 1], in two steps. For this purpose, we use properties 8 and 9 of the zero-padding and lifting operations from Section 5.1.

In the ﬁrst step of the mapping, each k-family K = ([n], E) is decomposed ′ ′ ′ ′ into two k-families, K1 = ([n], E1) and K2 = ([n], E2) via a mapping Fn(k) → ′ ′ Fn(k) × Fn(k) , such that K1 contains all the hyperedges including vn and K2 contains all the other hyperedges.

In the second step of the mapping, a (k)-family K2 = ([n − 1], E1) is obtained ′ by deleting vertex vn from the vertex set of K2. This mapping is a reduction from Fn(k) to Fn−1(k). Also a (k − 1)-family K1 = ([n − 1], E1) is obtained by deleting ′ ′ vertex vn from the vertex set of K1 and consequently from all the hyperedges in E1. This mapping is a reduction from Fn(k) to Fn−1(k − 1). The result after applying both steps is ψF : Fn(k) → Fn−1(k − 1) × Fn−1(k) . Deﬁnition 5.3. Deﬁne the mapping ψF as follows:

ψF : Fn(k) → Fn−1(k − 1) × Fn−1(k) , where ψF (K) = (K1 × K2) as explained above.

Note: The second step of the mapping is similar to applying the reverse of zero- ′ ′ pad operation on k2 and applying the reverse of (k − 1)-lift operation on k1 (see properties 8 and 9 from Section 5.1). 38 February 9, 2011

′ ′ ′ We call K1 and K2 the immediate elements , K1 = starvn (K) the star of vertex ′ n in K, K2 the star complement in K, K1 = linkvn (K) the link of vertex n in K, ψF and K2 the link complement in K. We use the notation K −−→ (K1 × K2) to show that ψF maps K to the pair of K1 and K2.

Example 5.4. For the 3-family in Example 1.2 Original Hypergraph: K = ([6], E) E = {123, 125, 136, 456} ′ ′ ′ Star of vertex n: K1 = ([6], E1) E1 = {136, 456} ′ ′ ′ Star Complement: K2 = ([6], E2) E2 = {123, 125} Link of vertex n: K1 = ([5], E1) E1 = {13, 45}. Link Complement: K2 = ([5], E2) E2 = {123, 125} Degree sequences for these hypergraphs are: d(K) = dK = (3, 2, 2, 1, 2, 2) ′ d(K′ ) = dK1 = (1, 0, 1, 1, 1, 2) 1 ′ ′ K2 d(K2) = d = (2, 2, 1, 0, 1, 0) K1 d(K1) = d = (1, 0, 1, 1, 1). K2 d(K2) = d = (2, 2, 1, 0, 1)

For any decomposition of K, using ψF , it is easy to prove the following properties: ′ ′ (1) E = E1 ∪˙ E2 ′ (2) E = E2 2 ′ (3) dK1 = Lk(dK1 ) ′ (4) dK2 = Z(dK2 ) ′ (5) e ∈ E1 ⇐⇒ (vn ∈ e) and (e ∈ E) ′ (6) e ∈ E1 ⇐⇒ (vn ∈/ e) and (e ∪ vn ∈ E1) ′ (7) e ∈ E2 ⇐⇒ (vn ∈/ e) and (e ∈ E) (8) e ∈ E2 ⇐⇒ (vn ∈/ e) and (e ∈ E) (9) ′ ′ dK = dK1 + dK2 n−1 K1 K2 K1 = d + d ; di /(k − 1) i=1 X′ K1 K1 K2 = d ; d (vi)/(k − 1) + (d ; 0) i∈[n−1] X = L(dK1 )+ Z(dK2 ) (5.1) ,where add is performed component-wise.

Proposition 5.5. The mapping ψF as introduced in Deﬁnition 5.3 is a bijection from Fn(k) to Fn−1(k − 1) × Fn−1(k).

Proof. We need to show that ψF is both one-to-one (injective) and onto (surjective).

• ψF is one-to-one: [proof by contradiction] Assume ψF maps two k-families K = ([n], A) ∈ Fn(k) and H = ([n],B) ∈ Fn(k) to the same pair K1 = ([n − 1], E1) ∈ Fn−1(k − 1) and K2 = PROPERTIES OF DEGREE SEQUENCES OF k-UNIFORM HYPERGRAPHS 39

([n − 1], E2) ∈ Fn−1(k). If K and H are not the same hypergraph, there exist at least one hyperedge, which is exactly in one of A or B. Assume e to be such a hyperedge. There are two possible cases for e with regard to containing vn. ′ 1. If vn ∈/ e and (e ∈ A or e ∈ B), then e ∈ E2 and therefore e ∈ A and e ∈ B. (Using property (7) of ψF .) ′ 2. If vn ∈ e and (e ∈ A or e ∈ B), then e ∈ E1 and therefore e ∈ A and e ∈ B. (Using property (5) of ψF .) Both cases possible result in contradictions. So ψF is one-to-one.

• ψF is onto: For any pair of hypergraphs K1 ∈ Fn−1(k − 1) and K2 ∈ Fn−1(k) we can apply the reverse of ψF to get K = ([n], E) ∈ Fn(k). First add vn to the vertex set and all of hyperedges of K1 to get the star of vertex vn in K, ′ ′ K1 = ([n], E1) ∈ Fn(k). Then add vn to the vertex set of K2 in order to ′ ′ ′ ′ form K2 = ([n], E2) ∈ Fn(k). Finally, set K to be ([n], E1∪˙ E2) ∈ Fn(k). ′ Note that since all hyperedges of K1 ∈ Fn(k) contain vn but none of the ′ ′ ′ hyperedges of K2 ∈ Fn(k) does, K1 and K2 have no hyperedge in common.

We extend the concept of the bijection ψF : Fn(k) → Fn−1(k − 1) × Fn−1(k) to a relation ψ : DS (k) → DS (k − 1) × DS (k) . DS n n−1 n−1 Deﬁnition 5.6. Deﬁne the relation ψDS as follows:

ψDS : DSn(k) → DSn−1(k − 1) × DSn−1(k)

ψDS associates a point d ∈ DSn(k) to the pair of points d1 ∈ DSn−1(k − 1) and d2 ∈ DSn−1(k) if there exist K, K1 and K2, respectively realizations of d, d1, and ψF d2 such that K −−→ (K1 × K2).

K K2 In other words, for each d ∈ DSn(k), d(K1) ∈ DSn−1(k − 1) and d ∈ DSn−1(k),

K ψDS K1 K2 ψF d −−−→ (d × d ) ⇔ K −−→ (K1 × K2) . Using equation (5.1), we have 

n−1 K ψDS K1 K2 K K1 K2 K1 d −−−→ (d × d ) ⇔ d = d + d ; di /(k − 1) . i=1 X Note 1: K, K1 and K2, do not need to be the unique realizations of d, d1, and d2.

Note 2: This relation is not a function. However, the reverse of the relation, i.e, −1 ψDS : DSn−1(k − 1) × DSn−1(k) → DSn(k) is a function.

−1 Note 3: Both relations ψDS and ψDS are onto. 40 February 9, 2011

It is easy to generalize the concept of the relation ψDS : DSn(k) → DSn−1(k − 1) × DS (k) to a relation ψ : D (k) → D (k − 1) × D (k) . n−1 D n n−1 n−1 Deﬁnition 5.7. Deﬁne the relation ψD as follows:

ψD : Dn(k) → Dn−1(k − 1) × Dn−1(k) for each set of point p ∈ Dn(k), p1 ∈ DSn−1(k − 1) and p2 ∈ Dn−1(k), n−1 ψD p −−→ (p1 × p2) ⇔ p1+ p2 ; p1i/(k − 1) . i=1 X This implies that Dn(k) is the Minkowski sum of zero-padded image of Dn−1(k) and lifted image of Dn−1(k − 1), i.e. k−1 Dn(k)= Z(Dn−1(k)) ⊕ L Dn−1(k − 1)

= ZDn−1(k) ⊕ LDn−1(k − 1). (5.2)

Therefore,

Theorem 5.8. There is no hole in Dn(k) if and only if there is no hole in ZDn−1(k) ⊕ LDn−1(k − 1).

Consequently, Problem 1.12 could be redused to the following problem:

Problem 5.9. Are there any lattice points in Dn(k) that can not be written as the summation of a pair of lattice points one in ZDn−1(k) and another one in LDn−1(k − 1)?

5.3. Some Interesting Properties of the Threshold Hypergraphs. In this section we introduce and prove some interesting properties of the threshold hypergraphs under the ψF bijection. These properties are also true for the extreme point of the polytope of hypergraphic sequences and the ψD relation.

Recall from Deﬁnition 4.5 that a k-hypergraph H = ([n], E) is called a “k- threshold” if there exists an n-tuple c = (c1,c2,...,cn) of real numbers such that n for all X ∈ k , we have X ∈ E if and only if c(X)= i∈X ci > 0. In the following proofs we look at X as a n-dimensional (0 − 1)-vector, (x1,...,xn), where xi = 1 P if and only if vi ∈ X.

K Proposition 5.10. A (lattice) point d ∈ Dn(k) is a k-threshold sequence, if and K only if Z(d ) ∈ Dn+1(k) is a k-threshold sequence.

Proof. (only if:) Consider K = ([n], E) to be the k-threshold hypergraph realizing K d . Assume c = (c1,c2,...,cn) is a n-tuple of real numbers for which for all n X ∈ k , we have X ∈ E if and only if c(X)= i∈X ci > 0. Claim: K′ = ([n + 1], E′) is the k-threshold hypergraph realizing Z(dK ), where P PROPERTIES OF DEGREE SEQUENCES OF k-UNIFORM HYPERGRAPHS 41

′ ′ ′ ′ n c = (c1,c2,..., cn+1) = (c1,c2,...,cn, (− i=1 |ci|) − 1) is a (n + 1)-tuple of real ′ n+1 ′ ′ ′ ′ numbers for which for all X ∈ k , we have X ∈ E if and only if c (X ) = ′ P ′ c > 0. i∈X i Proof of the claim: We need to show that P ′ ′ n+1 (1) Vertex vn+1 is not in any hyperedges of K , i.e. for all X ∈ k if ′ n ′ ′ X = (X;1) (for some X ∈ k−1 ), then X ∈/ E : This is true because ′ n ′ ′ ′ cn+1 = −( i=1 |ci|) − 1, therefore for any X containing vn, c (X ) = ′ ′ c ≤−1 < 0. i∈X i P (2) All other hyperedges stay the same, i.e. for all X′ ∈ n+1 if X′ = (X; 0) P k (for some X ∈ n ), then X ∈ E if and only if X′ ∈ E′: This is true because k if X′ = (X; 0), then ′ ′ c(X)= ci > 0 ⇐⇒ c(X )= ci > 0. ′ iX∈X iX∈X K ′ ′ (if:) If Z(d ) ∈ Dn+1(k) is a k-threshold sequence. Consider K = ([n + 1], E ) K ′ ′ ′ ′ to be the k-threshold hypergraph realizing Z(d ). Assume c = (c1,c2,...,cn+1) ′ (n+1) is a (n + 1)-tuple of real numbers for which for all X ∈ k . K Claim: K = ([n], E) is the k-threshold hypergraph realizing d , where c = (c1,c2,..., ′ ′ ′ n cn) = (c1,c2,...,cn)isa n-tuple of real numbers for which for all X ∈ k , we have X ∈ E if and only if c(X)= c > 0. i∈X i Proof of the claim: The claim is true because for all X′ ∈ E′ there exist an X ∈ n P k such that X′ = (X; 0), and ′ ′ c(X )= ci > 0 ⇐⇒ c(X)= ci > 0. ′ iX∈X iX∈X Therefore, X ∈ E if and only if X′ = (X; 0) ∈ E′.

Recall from property A.1 in Section 4.2 (Theorem 2.5 in [23]) that a degree sequence is an extreme point of Dn(k) if and only if it is a k-threshold sequence.

K Corollary 5.11. A (lattice) point d is an extreme point of Dn(k), if and only if K Z(d ) is an extreme point of Dn+1(k).

K Proposition 5.12. A (lattice) point d ∈ Dn(k) is a k-threshold sequence, if and k K only if L (d ) ∈ Dn+1(k + 1) is a (k + 1)-threshold sequence.

Proof. (only if:) Consider K = ([n], E) to be the k-threshold hypergraph realizing K d . Assume c = (c1,c2,...,cn) is a n-tuple of real numbers for which for all n X ∈ k , we have X ∈ E if and only if c(X)= i∈X ci > 0. n ci Let cmax = |max {ci}| then −1 ≤ ≤ 1. i=1 cmax P Claim: K′ = ([n + 1], E′) is the (k + 1)-threshold hypergraph realizing Lk(dK ), where c′ = (c′ ,c′ ,..., c′ ) = ( c1 − 1, c2 − 1,..., cn − 1, k) is a (n + 1)- 1 2 n+1 cmax cmax cmax ′ n+1 ′ ′ tuple of real numbers for which for all X ∈ k+1 , we have X ∈ E if and only if ′ ′ ′ c (X )= ′ c > 0. i∈X i Proof of the claim: We need to show that P 42 February 9, 2011

′ ′ ′ (1) Vertex vn+1 is in all hyperedges of K , i.e. for all X ∈ E there exist n ′ X ∈ k such that X = (X; 1): This is true because, for all i ≤ n we have ′ ci ′ n+1 set c = − 1. Therefore, for all X ∈ not containing vn+1 we i cmax k+1 have ′ ′ ci C(X )= ci = ( − 1) ≤ (1 − 1)=0. ′ ′ cmax ′ iX∈X iX∈X iX∈X (2) All other hyperedges stay the same except for vn+1 included in them, i.e. ′ n+1 ′ ′ ′ for all X ∈ k+1 if X = (X; 1), then X ∈ E if and only if X ∈ E : This is true because if X′ = (X; 1), then ′ ′ ci ci C(X )= ci = k + ( − 1) = k − k + ( ). ′ cmax cmax iX∈X iX∈X iX∈X Therefore,

′ i∈X ci C(X )= > 0 ⇔ C(X)= ci > 0. cmax P i∈X X k K ′ (if:) If L (d ) ∈ Dn+1(k +1) is a (k + 1)-threshold sequence. Consider K = ([n + 1], E′) to be the (k + 1)-threshold hypergraph realizing Lk(dK ). Assume ′ ′ ′ ′ ′ (n+1) c = (c1,c2,...,cn+1) is a (n+1)-tuple of real numbers for which for all X ∈ k . K Claim: K = ([n], E) is the k-threshold hypergraph realizing d , where c = (c1,c2,..., ′ ′ ′ ′ cn+1 ′ cn+1 ′ cn+1 cn) = (c1 + k ,c2 + k ,...,cn + k )isa n-tuple of real numbers for which for n all X ∈ k , we have X ∈ E if and only if c(X)= i∈X ci > 0. Proof of the claim: The claim is true because for all X′ ∈ E′ there exist an X ∈ n P k such that X′ = (X; 1), and ′ ′ ′ ′ ′ ′ cn+1 C(X )= ci = cn+1 + ci = (ci + )= ci. ′ k iX∈X iX∈X iX∈X iX∈X Therefore, X ∈ E if and only if X′ = (X; 1) ∈ E′.

K Corollary 5.13. A (lattice) point d is an extreme point of Dn(k), if and only if k K L (d ) is an extreme point of Dn+1(k + 1).

Proposition 5.14. The bijection ψF maps a threshold hypergraph K ∈ Fn(k) into a pair of threshold hypergraphs K1 ∈ Fn−1(k − 1) and K2 ∈ Fn−1(k).

Proof. Consider K = ([n], E). Assume c = (c1,c2,...,cn) is a n-tuple of real n numbers for which for all X ∈ k , we have X ∈ E if and only if c(X)= i∈X ci > 0. ′ ′ P It is enough to show that K1 and K2 as explained for the mapping ψF are threshold hypergraphs. Using propositions 5.10 and 5.12, it follows that K1 and K2 are also threshold hypergraphs. ′ ′ ′ ′ ′ Claim 1: K2 = ([n], E2) is the k-threshold hypergraph, where c = (c1,c2,..., ′ n−1 cn) = (c1,c2,...,cn−1, (− i=1 |ci|) − 1) is a n-tuple of real numbers for which for ′ n ′ ′ ′ ′ ′ all X ∈ k , we have X ∈ E if and only if c (X )= i∈X′ ci > 0. n−1 P ci Let cmax = |max {ci}| then for 1 ≤ i ≤ n − 1 we have −1 ≤ ≤ 1. i=1 P cmax PROPERTIES OF DEGREE SEQUENCES OF k-UNIFORM HYPERGRAPHS 43

′ ′ ′ ′ ′ Claim 2: K1 = ([n], E1) is the (k)-threshold hypergraph, where c = (c1,c2,..., c′ ) = ( c1 − 1, c2 − 1,..., cn−1 − 1, k − 1) is a n-tuple of real numbers for which n cmax cmax cmax ′ n ′ ′ ′ ′ ′ for all X ∈ k , we have X ∈ E1 if and only if c (X )= i∈X′ ci > 0. The proof of claim 1 in this proof is similar to the proof of the claim in the only if part of the proof of Proposition 5.10 and the proof ofP claim 2 in this proof is similar to the proof of the claim in the only if part of the proof of Proposition 5.12. However, for the purpose of completeness we include them here. Proof of the claim 1: We need to show that ′ n−1 ′ (1) Vertex vn is not in any hyperedges of K2, i.e. for all X ∈ k−1 if X = ′ ′ ′ n−1 (X; 1), then X ∈/ E : This is true because cn = −( i=1 |ci|) − 1, therefore ′ ′ ′ ′ for any X containing vn, c (X )= ′ c ≤−1 < 0. i∈X i P (2) All other hyperedges stay the same, i.e. for all X ∈ n−1 if X′ = (X; 0), P k then X ∈ E if and only if X′ ∈ E′ : This is true because if X′ = (X; 0), 2 then ′ ′ c(X)= ci > 0 ⇐⇒ c(X )= ci > 0. ′ iX∈X iX∈X Proof of the claim 2: We need to show that ′ ′ ′ (1) Vertex vn is in all hyperedges of K1, i.e. for all X ∈ E there exist X ∈ n−1 ′ k−1 such that X = (X; 1): This is true because, for all i ≤ n − 1 we have ′ ci ′ n set c = − 1. Therefore, for all X ∈ not containing vn+1 we have i cmax k ′ ′ ci C(X )= ci = ( − 1) ≤ (1 − 1)=0. ′ ′ cmax ′ iX∈X iX∈X iX∈X ′ n (2) All possible hyperedges containing vn stay the same, i.e. for all X ∈ k if X′ = (X; 1), then X ∈ E if and only if X′ ∈ E′ : This is true because if 1 X′ = (X; 1), then ′ ′ ci C(X )= ci = (2k +1)+ ( − 1) ′ cmax iX∈X iX∈X ≥ (2k +1)+ (−1 − 1)=1 > 0. ′ iX∈X ′ ′ ci ci C(X )= ci = k − 1+ ( − 1)=(k − 1) − (k − 1) + ( ). ′ cmax cmax iX∈X iX∈X iX∈X Therefore,

′ i∈X ci C(X )= > 0 ⇔ C(X)= ci > 0. cmax P i∈X X

Note: The reverse of this proposition is not necessarily true. 44 February 9, 2011

Appendix A. Preliminary Steps for Theorem 3.8 In this section we explain the simplified cases that motivated us to formulate Theorem 3.8. Just as the first step in the majority of attempts to characterize degree sequences, which is dividing the vertices into different subsets, we divide vertices of a nonincreasing sequence of nonnegative integers ϕ (and any k-family H representing it if any) into three partitions: S : the set of the s highest-degree vertices. S = {v1, v2, ..., vs} = [s] T : the set of the t lowest-degree vertices. T = {vn−t+1, vn−t+2, ..., vn} = [n − t +1,n] B : the rest of the vertices. |B| = n − s − t = b. B = {vs+1, vs+2, ..., vn−t} = [s +1,n − t]

We will start with the simple case, S = {v1},B = {v2, v3, ..., vn},T = ∅, and investigate the constraints , i.e. the necessary conditions, for a sequence to be k-hypergraphic. Step by step, we make the case more general and add to the list of constraints until we reach the general case, S = {v1, v2, ..., vs},T = {vn−t+1, vn−t+2, ..., vn}, and B = {vs+1, vs+2, ..., vn−t}. The general case has been studied in Section 3.2 in detail so we do not investigate it here.

For each case we try to obtain an upperbounds on the sum of degrees of vertices in S, j∈S dj , based on the sum of degrees of vertices in T , j∈T dj , and the number of vertices in S,T and B. As we previously said in Section 3, hyperedges that contributeP the most to the degree of vertices in S and use theP least from the degree of vertices in T play a more signiﬁcant role in determining these upperbounds. Therefore, those are the hyperedges we consider ﬁrst.

The following are the diﬀerent cases:

(1) S = {v1}:

(a) S = {v1},B = {v2, v3, ..., vn}, and T = ∅: The maximum degree in a k-family H on [n], happens when H contains all the possible k-hyperedges connecting v1 and any of the diﬀerent combinations of k − 1 vertices chosen from the rest of the vertices. Therefore, n − 1 d ≤ . (A.1) 1 k − 1 See Table 1 for the result of Theorem 3.8 for this case.

(b) S = {v1},B = {v2, v3, ..., vn−1}, and T = {vn}: Considering the degree of the low-degree vertices might put a bound on the maximum degree. For inequality(A.1) to be an equality any vertices vi=16 should appears at least in all possible hyperedges connecting v1 and vi, with any combination of k − 2 other vertices. Therefore, n−2 n−2 each vi should appears in at least k−2 hyperedges. So if dn < k−2 PROPERTIES OF DEGREE SEQUENCES OF k-UNIFORM HYPERGRAPHS 45

Table 1. case 1.a. S = {v1}, B = {v2, ..., vn}, and T = ∅

Mi αi γi 1 n−1 0 M0 = M(1, 0) = 1 k−1 0 1 0 M = M(0, 0) = n−1 0 0 1 k−1 Constraint: n − 1 n − 1 d ≤ α M + α M =1. +0= 1 0 0 1 1 k − 1 k − 1 

Table 2. case 1.b. S = {v1}, B = {v2, ..., vn−1}, and T = {vn}

Mi αi γi 1 n−2 1 M0 = M(1, 0) = 1 k−1 0 1 0 M = M(1, 1) = 1 n−2 1 1 1 1 1k−21 M = M(0,γ)= n−1−γ 0 0 1

, we have a tighter restriction on d1. n − 2 d ≤ + d . (A.2) 1 k − 1 n The right hand side of the inequality(A.2) divides the hyperedges containing v1 into two partitions, those which do not contain vn and those which contain vn. The right side inequality is strict when vn occurs in hyperedges without v1. n−2 n−2 n−1 If dn ≥ k−2 then k−1 + dn ≥ k−1 . Therefore, inequality(A.2) does not result in a tighter constrain than inequality (A.1). This happens when the number of vertices, other than v1 and vn, is too small to provide dn diﬀerent combinations of k − 2 vertices, needed to form hyperedges with v1 and vn.

In the general case, n − 2 n − 2 d ≤ + min{d , }. (A.3) 1 k − 1 n k − 2 However, for the purpose of simplicity we will keep the set of constraints in the inequality (A.3) as two separate sets of constraints presented in inequalities (A.1) and (A.2). See Table 2 for the result of Theorem 3.8 for this case.

(c) S = {v1},B = {v2, v3, ..., vn−t}, and T = {vn−t−1, ..., vn}: We can get more restricted constraints by looking at the degrees of 46 February 9, 2011

Table 3. case 1.c. S = {v1}, B = {v2, ..., vn−t}, and T = {vn−t+1,...,vn}

Mi αi γi 1 n−t−1 t M0 = M(1, 0) = 1 k−1 0 1 0 M = M(1, 1) = 1 n−t−1 t 1 1 1 1 k−2 1 M = M(1, 2) = 1 n−t−1 t 1 2 2 1 k−3 2 M = M(1,i)= 1 n−t−1 t 1 i 1≤i

vi∈T i=0 X X z z d1 ≤ Mi.αi + ( di − i.Mi)αz+1/γz+1. i=0 i=1 X vXi∈T X

some of the least-degree vertices. n − t − 1 d ≤ + d (A.4) 1 k − 1 i vXi∈T The right hand side of the inequality(A.4) divides the hyperedges containing v1 into two partitions, those which do not contain any of the last t vertices and those which contain exactly one of the last t vertices. We are not considering hyperedges that contain two or more of the last t vertices, because by replacing such a hyperedge with several hyperedges, each contains exactly one of the last t vertices, we can have more hyperedges containing v1. To be able to do this replacing n−t−1 we need dn−t−1 ≤ k−2 , i.e. for all y ∈ [n − t − 1,n] the remaining n − t − 1 vertices be able to provide d diﬀerent combinations of y k − 2 vertices to form k-hyperedges with v1 and dy. This condition is true for all dy, y ∈ [n − t − 1,n], if it is true for the largest one, i.e. n−t−1 dn−t−1. The constraints for the case dn−t−1 > k−2 , are tighter than inequality (A.4). See Table 3 for constraints on more cases and the result of Theorem 3.8.

(2) S = {v1, v2}: Including the next highest degree vertex in the inequalities can also tighten the constraints. PROPERTIES OF DEGREE SEQUENCES OF k-UNIFORM HYPERGRAPHS 47

Table 4. case 2.a. S = {v1, v2}, B = {v3, ..., vn}, and T = ∅

Mi αi γi 2 n−2 0 M0 = M(2, 0) = 2 k−2 0 2 0 M = M(1, 0) = 2 n−2 0 1 0 1 1k−10 M = M(0, 0) = n−2 0 0 2 k−1 Constraint: n − 2 2 n − 2 n − 1 d + d ≤ 2. +1. +0=2 1 2 k − 2 1 k − 1 k − 1

(a) S = {v1, v2},B = {v3, ..., vn}, and T = ∅: First let us include d2 in inequality(A.1). n − 2 n − 2 n − 1 d + d ≤ 2 +2 =2 . (A.5) 1 2 k − 1 k − 2 k − 1 Inequality(A.5) divides the hyperedges containing v1 and/or v2 into two partitions, those which contain only one and those which contain n−2 both. k−1 of the edges contain only v1, the same number contain only v , and n−2 contain both, which should be counted twice be- 2 k−2 cause they contribute to both d and d . However, since we assumed 1 2 ϕ is not increasing, inequality (A.5) is not a new constrain from our previous result, inequality(A.1). See Table 4 for the result of Theorem 3.8 in this case.

(b) S = {v1, v2},B = {v3, ..., vn−1}, and T = {vn}: Now let us separate vn in inequality(A.5). n − 3 n − 3 n − 2 d + d ≤ 2 + 2( + d )=2 +2d . (A.6) 1 2 k − 1 k − 2 n k − 1 n Inequality (A.6) divides hyperedges containing both v1 and v2 into two partitions those which do not contain vn and those which contain vn. To let dn contribute the most to the sum d1 + d2, we assume each hyperedge that contains vn, also contains both v1 and v2. For the n−2 same reason explained in Step (1c), if dn < k−2 , inequality(A.6) is a tighter restriction than inequality(A.5), but this is still the same result as inequality (A.2).

n−3 If k−3 < dn , since n−3 vertices other than v1, v2 and vn can provide only n−3 distinct combinations to form hyperedges with v , v and k−3 1 2 v , at most n−3 of d hyperedges containing v can contribute 2 to n k−3 n n the sum d + d and the rest can contribute at most 1. 1 2 n − 2 n − 3 n − 3 d + d ≤ 2 +2 + d − ⇒ 1 2 k − 1 k − 3 n k − 3 n − 2 n − 3 d + d ≤ 2 + + d . (A.7) 1 2 k − 1 k − 3 n 48 February 9, 2011

Table 5. case 2.b. S = {v1, v2}, B = {v2, ..., vn−1}, and T = {vn}

Mi αi γi 2 n−3 1 M0 = M(2, 0) = 2 k−2 0 2 0 M = M(1, 0) = 2 n−3 1 1 0 1 1k−10 M = M(2, 1) = 2 n−3 1 2 1 2 2k−31 M = M(1, 1) = 2 n−3 1 1 1 3 1k−21 M = M(0,γ)= n−3−γ 1 0 γ 3

vi∈T i=0 X X z z d1 + d2 ≤ Mi.αi + ( di − i.Mi)αz+1/γz+1. i=0 i=1 X vXi∈T X

n−3 Therefore, if k−3 < dn , then inequality (A.7) is an additional constrain on d. Note that for the case n−2 < d , as it was explained in k−2 n item (1b), there are some edges that contain v but do not contain v n 1 nor v2. Therefore, they do not contribute to the sum. See Table 5 for the result of Theorem 3.8 for this case.

(c) S = {v1, v2},B = {v3, ..., vn−t}, and T = {vn−t+1,...,vn}: We do not need this case in order to build our argument. See Table 6 for the result of Theorem 3.8 for this case.

(3) S = {v1, v2, ..., vs}: Including other high degree vertices might also tighten the inequality.

(a) S = {v1, v2, ..., vs},B = {vs+1, ..., vn}, and T = ∅: See Table 7 for the result of Theorem 3.8 for this case.

(b) S = {v1, v2, ..., vs},B = {vs+1, ..., vn−1}, and T = {vn}:

s s n − s − 1 d ≤ sd + y . (A.8) i n y k − y y=1 vXi∈S X PROPERTIES OF DEGREE SEQUENCES OF k-UNIFORM HYPERGRAPHS 49

Table 6. case 2.c. S = {v1, v2}, B = {v3, ..., vn−t}, and T = {vn−t+1,...,vn}

Mi αi γi 2 n−t−2 t M0 = M(2, 0) = 2 k−2 0 2 0 M = M(1, 0) = 2 n−t−2 t 1 0 1 1 k−1 0 M = M(2, 1) = 2 n−t−2 t 2 1 2 2 k−3 1 M = M(1, 1) = 2 n−t−2 t 1 1 3 1 k−2 1 M = M(2, 2) = 2 n−t−2 t 2 2 4 2 k−4 2 M = M(2, 3) = 2 n−t−2 t 2 3 4 2 k−5 3 M = M(1, 2) = 2 n−t−2 t 1 2 5 1 k−3 2 . . n−t−2−γ 1 2(t+1)

+(2/3). di−)0M0 +0M1 +1M2 +1M3 +2M4)

vi∈T . X . z iv) if dn ≥ Mi.γi,

vi∈T i=0 X X z z d1 + d2 ≤ Mi.αi + ( di − i.Mi)αz+1/γz+1. i=0 i=1 X vXi∈T X

Inequality (A.8) divides hyperedges into two major partitions, those which contain vn and those which do not contain vn. Each of the hyperedges containing vn can contribute up to one to the degree of each vi ∈ S. Those which do not contain vn may contain any number y (≤ s, k) of the highest-degree vertices and k − y of the vertices in B. Each of the later kind of hyperedges contributes y to the sum

vi∈S di. P 50 February 9, 2011

Table 7. case 3.a. S = {v1,...vs}, B = {vs+1, ..., vn}, and T = ∅

Mi αi γi s n−s 0 M0 = M(s, 0) = s 0 0 s 0 s n−s 0 M1 = M(s − 1, 0) = s−1 1 0 s-1 0 . . s n−s 0 Ms+1 = M(0, 0) = 0 k 0 0 0 Constraint: min(k,s) min(k,s) s n − s s − 1 n − s d ≤ i = s i i k − i i − 1 k − i vi∈S i=1 i=1 X X min(k,s) X s − 1 n − s n − 1 = s( )= s. i − 1 k − i k − 1 i=1 n − 1X ⇒ d ≤ s. i k − 1 vXi∈S

We know that, s s s n − s − 1 s − 1 n − s − 1 y = s y k − y y − 1 k − y y=1 y=1 X X s s − 1 n − s − 1 = s y − 1 k − y y=1 X s ′ ′ ′ ′ s n − s − 2 (assuming s = s − 1,y = y − 1) = s ′ ′ ′ y k − y − 1 yX=0 n − 2 = s (A.9) k − 1 Therefore,

s n − 2 d ≤ s(d + ). (A.10) y n k − 1 y=1 X which is again not a new result from inequality(A.2). Now we use the generalization of the the same idea we used for inequality(A.7) in order to gain some additional constraints on the degree sequence d.

In order for vn to contribute the most to the degree of vertices in S, it should s n−s−1 1. Form all possible s k−s−1 hyperedges that contain all s vertices in S and k − s − 1 vertices in B. Each of these hyperedges s contribute s to the sum y=1 dy. 2. Form all possible s n−s−1 hyperedges that contain s − 1 s−1 Pk−s vertices in S and k − s vertices in B. Each of these hyperedges s contribute s − 1 to the sum y=1 dy. P PROPERTIES OF DEGREE SEQUENCES OF k-UNIFORM HYPERGRAPHS 51

s n−s−1 3. Form all possible s−2 k−s+1 hyperedges that contain s − 2 vertices in S and k−s+1 vertices in B. Each of these hyperedges s contribute s − 2 to the sum y=1 dy. ... s n−sP−1 z+1. Form all possible s−z k−s+z−1 hyperedges that contain s − z vertices in S and k − s + z − 1 vertices in B. Each of these s hyperedges contribute s − z to the sum y=1 dy. z+2. Some of the hyperedges that contain s − z − 1 vertices in S and k − s + z vertices in B. Each of theseP hyperedges contribute s s − z − 1 to the sum y=1 dy.

We can determine the valueP of z based on the three possibilities for dn. s n−s−1 1. If dn < s k−s−1 , set z = 0. 2. If d = s ( s n−s−1 )= n−1 , set z = s. n i=0 s−i k−s−1+i k−1 3. Otherwise, set z such that 0 ≤ z

(c) S = {v1, v2, ...vs},B = {vs +1, ..., vn−t}, and T = {vn−t−1, ..., vn}: This case has been studied in details in Section 3.2. However, in Section 3.2 to be consistant to the Koren’s notation in [21], we did not give a speciﬁc name to set B = ([n] − S − T ). 52 February 9, 2011

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~~Department of Computer Science, The University of Chicago~~