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 inter- ested 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 dif- ferent 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 hyper- graphic 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 prob- lems 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 proper- ties 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 con- cerning 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 (hy- per)edges incident with the vertex v. The (ordered) degree sequence of a (hy- per)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 rep- resentation of H. 1.2. The Characterization Problem for Hypergraphic Sequences. A non- negative integer sequence d = (d1, d2, .., dn) is called graphic if there exists a (sim- ple) 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 characteriza- tions 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.