Complete K-uniform Hypergraph Research Articles

Ramsey numbers of hypergraphs with a given size

Abstract The q-colour Ramsey number of a k-uniform hypergraph H is the minimum integer N such that any q-colouring of the complete k-uniform hypergraph on N vertices contains a monochromatic copy of H. The study of these numbers is one of the central topics in Combinatorics. In 1973, Erdős and Graham asked to maximise the Ramsey number of a graph as a function of the number of its edges. Motivated by this problem, we study the analogous question for hypergaphs. For fixed $k \ge 3$ and $q \ge 2$ we prove that the largest possible q-colour Ramsey number of a k-uniform hypergraph with m edges is at most $\mathrm{tw}_k(O(\sqrt{m})),$ where tw denotes the tower function. We also present a construction showing that this bound is tight for $q \ge 4$ . This resolves a problem by Conlon, Fox and Sudakov. They previously proved the upper bound for $k \geq 4$ and the lower bound for $k=3$ . Although in the graph case the tightness follows simply by considering a clique of appropriate size, for higher uniformities the construction is rather involved and is obtained by using paths in expander graphs.

The growth rate of multicolor Ramsey numbers of 3-graphs

The q-color Ramsey number of a k-uniform hypergraph G, denoted r(G; q), is the minimum integer N such that any coloring of the edges of the complete k-uniform hypergraph on N vertices contains a monochromatic copy of G. The study of these numbers is one of the most central topics in combinatorics. One natural question, which for triangles goes back to the work of Schur in 1916, is to determine the behavior of r(G; q) for fixed G and q tending to infinity. In this paper, we study this problem for 3-uniform hypergraphs and determine the tower height of r(G; q) as a function of q. More precisely, given a hypergraph G, we determine when r(G; q) behaves polynomially, exponentially or double exponentially in q. This answers a question of Axenovich, Gyárfás, Liu and Mubayi.

A Measure for the Vulnerability of Uniform Hypergraph Networks: Scattering Number

The scattering number of a graph G is defined as s(G)=max{ω(G−X)−|X|:X⊂V(G),ω(G−X)>1}, where X is a cut set of G, and ω(G−X) denotes the number of components in G−X, which can be used to measure the vulnerability of network G. In this paper, we generalize this parameter to a hypergraph to measure the vulnerability of uniform hypergraph networks. Firstly, some bounds on the scattering number are given. Secondly, the relations of scattering number between a complete k-uniform hypergraph and complete bipartite k-uniform hypergraph are discussed.

K-Zero-Divisor and Ideal-Based k-Zero-Divisor Hypergraphs of Some Commutative Rings

Let R be a commutative ring with nonzero identity and k≥2 be a fixed integer. The k-zero-divisor hypergraph Hk(R) of R consists of the vertex set Z(R,k), the set of all k-zero-divisors of R, and the hyperedges of the form {a1,a2,a3,…,ak}, where a1,a2,a3,…,ak are k distinct elements in Z(R,k), which means (i) a1a2a3⋯ak=0 and (ii) the products of all elements of any (k−1) subsets of {a1,a2,a3,…,ak} are nonzero. This paper provides two commutative rings so that one of them induces a family of complete k-zero-divisor hypergraphs, while another induces a family of k-partite σ-zero-divisor hypergraphs, which illustrates unbalanced or asymmetric structure. Moreover, the diameter and the minimum length of all cycles or girth of the family of k-partite σ-zero-divisor hypergraphs are determined. In addition to a k-zero-divisor hypergraph, we provide the definition of an ideal-based k-zero-divisor hypergraph and some basic results on these hypergraphs concerning a complete k-partite k-uniform hypergraph, a complete k-uniform hypergraph, and a clique.

Defective colorings on k-uniform hypergraphs

In this article, we introduce the concepts of proper coloring, defective coloring and defective coloring without monochromatic cycle on hypergraphs together with their chromatic numbers. Properties and relationship among these three types of coloring on hypergraphs are explored. Chromatic number of each type of coloring for several hypergraphs, espeacially, complete k-uniform hypergraph on n vertices and a complete r-partite k-uniform hypergraph are given. Lower bounds of the defective chromatic number of a hypergraph involving its clique number are provided.

Adjacency spectra of random and complete hypergraphs

We present progress on the problem of asymptotically describing the adjacency eigenvalues of random and complete uniform hypergraphs. There is a natural conjecture arising from analogy with random matrix theory that connects these spectra to that of the all-ones hypermatrix. Several of the ingredients along a possible path to this conjecture are established, and may be of independent interest in spectral hypergraph/hypermatrix theory. In particular, we provide a bound on the spectral radius of the symmetric Bernoulli hyperensemble, and show that the spectrum of the complete k-uniform hypergraph for k=2,3 is close to that of an appropriately scaled all-ones hypermatrix.

Simplifying the computation of the spectrum of the complete k-uniform hypergraph

In this paper, we give a method to simplify the computation of the spectrum of the complete k-uniform hypergraph with n vertices, which gives a partial answer to a question posed by Cooper and Dutle [1]. We show that the eigenvalues of a 3-uniform hypergraph with n vertices can be computed by the roots of a sequence of cubic polynomials. For a complete 4-uniform hypergraph with n vertices, all of its eigenvalues can be found by the roots of a sequence of fifth-degree polynomials and a sequence of eleventh-degree polynomials.

The List-Chromatic Number of Complete Multipartite Hypergraphs and Multiple Covers by Independent Sets

This paper deals with list colorings of uniform hypergraphs. Let H(m, r, k) be the complete r-partite k-uniform hypergraph with parts of equal size m in which each edge contains exactly one vertex from some k ≤ r parts. Using results on multiple covers by independent sets, we establish that, for fixed k and r, the list-chromatic number of H(m, r, k) is (1 + o(1)) logr/(r−k+1)(m) as m → ∞.

On multicolor Ramsey numbers for loose [formula omitted]-paths of length three

We show that there exists an absolute constant A such that for each k≥2 and every coloring of the edges of the complete k-uniform hypergraph on Ar vertices with r colors, r≥rk, one of the color classes contains a loose path of length three.

Monochromatic loose path partitions in [formula omitted]-uniform hypergraphs

A conjecture of Gyárfás and Sárközy says that in every 2-coloring of the edges of the complete k-uniform hypergraph Knk, there are two disjoint monochromatic loose paths of distinct colors such that they cover all but at most k−2 vertices. A weaker form of this conjecture with 2k−5 uncovered vertices instead of k−2 is proved. Thus the conjecture holds for k=3. The main result of this paper states that the conjecture is true for all k≥3.

Fractional clique decompositions of dense graphs and hypergraphs

Our main result is that every graph G on n≥104r3 vertices with minimum degree δ(G)≥(1−1/104r3/2)n has a fractional Kr-decomposition. Combining this result with recent work of Barber, Kühn, Lo and Osthus leads to the best known minimum degree thresholds for exact (non-fractional) F-decompositions for a wide class of graphs F (including large cliques). For general k-uniform hypergraphs, we give a short argument which shows that there exists a constant ck>0 such that every k-uniform hypergraph G on n vertices with minimum codegree at least (1−ck/r2k−1)n has a fractional Kr(k)-decomposition, where Kr(k) is the complete k-uniform hypergraph on r vertices. (Related fractional decomposition results for triangles have been obtained by Dross and for hypergraph cliques by Dukes as well as Yuster.) All the above new results involve purely combinatorial arguments. In particular, this yields a combinatorial proof of Wilson's theorem that every large F-divisible complete graph has an F-decomposition.

On the random greedy F-free hypergraph process

Let F be a strictly k-balanced k-uniform hypergraph with e(F)≥|F|−k+1 and maximum co-degree at least two. The random greedy F-free process constructs a maximal F-free hypergraph as follows. Consider a random ordering of the hyper-edges of the complete k-uniform hypergraph Knk on n vertices. Start with the empty hypergraph on n vertices. Successively consider the hyperedges e of Knk in the given ordering and add e to the existing hypergraph provided that e does not create a copy of F. We show that asymptotically almost surely this process terminates at a hypergraph with O˜(nk−(|F|−k)/(e(F)−1)) hyperedges. This is best possible up to logarithmic factors.

On induced Ramsey numbers fork-uniform hypergraphs

Let denote the complete k-uniform k-partite hypergraph with classes of size t and the complete k-uniform hypergraph of order s. One can show that the Ramsey number for and satisfies when t = so(1) as s →∞. The main part of this paper gives an analogous result for induced Ramsey numbers: Let be an arbitrary k-partite k-uniform hypergraph with classes of size t and an arbitrary k-graph of order s. We use the probabilistic method to show that the induced Ramsey number (i.e. the smallest n for which there exists a hypergraph such that any red/blue coloring of yields either an induced red copy of or an induced blue copy of ) satisfies . © 2015 Wiley Periodicals, Inc. Random Struct. Alg., 48, 5–20, 2016

Decomposing Complete 3-Uniform Hypergraph into 5-Cycles

About the Katona-Kierstead definition of a Hamiltonian cycles in a uniform hypergraph, a decomposition of complete k-uniform hypergraph Kn(k) into Hamiltonian cycles studied by Bailey-Stevens and Meszka-Rosa. For n≡2,4,5 (mod 6), we design algorithm for decomposing the complete 3-uniform hypergraphs into Hamiltonian cycles by using the method of edge-partition. A decomposition of Kn(3) into 5-cycles has been presented for all admissible n≤17, and for all n=4m +1, m is a positive integer. In general, the existence of a decomposition into 5-cycles remains open. In this paper, we use the method of edge-partition and cycle sequence proposed by Jirimutu and Wang. We find a decomposition of K20(3) into 5-cycles.

Ramsey Theory, integer partitions and a new proof of the Erdős–Szekeres Theorem

Let H be a k-uniform hypergraph whose vertices are the integers 1,…,N. We say that H contains a monotone path of length n if there are x1<x2<⋯<xn+k−1 so that H contains all n edges of the form {xi,xi+1,…,xi+k−1}. Let Nk(q,n) be the smallest integer N so that every q-coloring of the edges of the complete k-uniform hypergraph on N vertices contains a monochromatic monotone path of length n. While the study of Nk(q,n) for specific values of k and q goes back (implicitly) to the seminal 1935 paper of Erdős and Szekeres, the problem of bounding Nk(q,n) for arbitrary k and q was studied by Fox, Pach, Sudakov and Suk.Our main contribution here is a novel approach for bounding the Ramsey-type numbers Nk(q,n), based on establishing a surprisingly tight connection between them and the enumerative problem of counting high-dimensional integer partitions. Some of the concrete results we obtain using this approach are the following:•We show that for every fixed q we have N3(q,n)=2Θ(nq−1), thus resolving an open problem raised by Fox et al.•We show that for every k≥3, Nk(2,n)=2⋅⋅2(2−o(1))n where the height of the tower is k−2, thus resolving an open problem raised by Eliáš and Matoušek.•We give a new pigeonhole proof of the Erdős–Szekeres Theorem on cups-vs-caps, similar to Seidenberg's proof of the Erdős–Szekeres Lemma on increasing/decreasing subsequences.

Decompositions of complete uniform hypergraphs into Hamilton Berge cycles

In 1973 Bermond, Germa, Heydemann and Sotteau conjectured that if n divides (nk), then the complete k-uniform hypergraph on n vertices has a decomposition into Hamilton Berge cycles. Here a Berge cycle consists of an alternating sequence v1,e1,v2,…,vn,en of distinct vertices vi and distinct edges ei so that each ei contains vi and vi+1. So the divisibility condition is clearly necessary. In this note, we prove that the conjecture holds whenever k≥4 and n≥30. Our argument is based on the Kruskal–Katona theorem. The case when k=3 was already solved by Verrall, building on results of Bermond.

On cyclic Hamiltonian decompositions of complete [formula omitted]-uniform hypergraphs

A decomposition C={C1,C2,…,Ch} of the complete k-uniform hypergraph Knk of order n is called cyclic Hamiltonian if each Ci∈C, i∈{1,2,…,h}, is a Hamiltonian cycle in Knk and there exists a permutation σ of the vertex set of Knk having exactly one cycle in its cycle decomposition such that for every cycle Ci∈C its set of edges coincides with an orbit of 〈σ〉 when acting on the edge set of Knk. In this paper it is shown that Knk admits a cyclic Hamiltonian decomposition if and only if n and k are relatively prime and λ=min{d>1:d|n}>nk.

Extensions of Results on Rainbow Hamilton Cycles in Uniform Hypergraphs

Let $${K_n^{(k)}}$$ K n ( k ) be the complete k-uniform hypergraph, $${k\ge3}$$ k ? 3 , and let l be an integer such that 1 ≤ l ≤ k?1 and k?l divides n. An l-overlapping Hamilton cycle in $${K_n^{(k)}}$$ K n ( k ) is a spanning subhypergraph C of $${K_n^{(k)}}$$ K n ( k ) with n/(k?l) edges and such that for some cyclic ordering of the vertices each edge of C consists of k consecutive vertices and every pair of consecutive edges in C intersects in precisely l vertices. An edge-coloring of $${K_n^{(k)}}$$ K n ( k ) is (a, r)-bounded if every subset of a vertices of $${K_n^{(k)}}$$ K n ( k ) is contained in at most r edges of the same color. In this paper, we refine recent results of the first author, Frieze and Rucinski by proving that there is a constant c = c(k, l) such that every $${(\ell, cn^{k-\ell})}$$ ( l , c n k - l ) -bounded edge-colored $${K_n^{(k)}}$$ K n ( k ) in which no color appears more that cn k-1 times contains a rainbow l-overlapping Hamilton cycle. We also show that there is a constant c? = c?(k, l) such that every (l, c?n k-l)-bounded edge-colored $${K_n^{(k)}}$$ K n ( k ) contains a properly colored l-overlapping Hamilton cycle.

The coloring complex and cyclic coloring complex of a complete k-uniform hypergraph

In this paper, we study the homology of the coloring complex and the cyclic coloring complex of a complete k-uniform hypergraph. We show that the coloring complex of a complete k-uniform hypergraph is shellable, and we determine the rank of its unique nontrivial homology group in terms of its chromatic polynomial. We also show that the dimension of the (n−k−1)st homology group of the cyclic coloring complex of a complete k-uniform hypergraph is given by a binomial coefficient. Further, we discuss a complex whose r-faces consist of all ordered set partitions [B1,…,Br+2] where none of the Bi contain a hyperedge of the complete k-uniform hypergraph H and where 1∈B1. It is shown that the dimensions of the homology groups of this complex are given by binomial coefficients. As a consequence, this result gives the dimensions of the multilinear parts of the cyclic homology groups of C[x1,…,xn]/{xi1…xik|i1…ik is a hyperedge of H}.

The covering radius of extreme binary 2-surjective codes

The covering radius of binary 2-surjective codes of maximum length is studied in the paper. It is shown that any binary 2-surjective code of M codewords and of length $$n = {M-1 \choose \left\lfloor(M-2)/2\right\rfloor}$$ has covering radius $$\frac{n}{2} - 1$$ if M ? 1 is a power of 2, otherwise $$\left\lfloor\frac{n}{2}\right\rfloor$$ . Two different combinatorial proofs of this assertion were found by the author. The first proof, which is written in the paper, is based on an existence theorem for k-uniform hypergraphs where the degrees of its vertices are limited by a given upper bound. The second proof, which is omitted for the sake of conciseness, is based on Baranyai's theorem on l-factorization of a complete k-uniform hypergraph.