N-vertex Hypergraph Research Articles

Sum-distinguishing number of sparse hypergraphs

A vertex labeling of a hypergraph is sum distinguishing if it uses positive integers and the sums of labels taken over the distinct hyperedges are distinct. Let s(H) be the smallest integer N such that there is a sum-distinguishing labeling of H with each label at most N. The largest value of s(H) over all hypergraphs on n vertices and m hyperedges is denoted s(n,m). We prove that s(n,m) is almost-quadratic in m as long as m is not too large. More precisely, the following holds: If n≤m≤nO(1) then s(n,m)=m2w(m),where w(m) is a function that goes to infinity and is smaller than any polynomial in m.The parameter s(n,m) has close connections to several other graph and hypergraph functions, such as the irregularity strength of hypergraphs. Our result has several applications, notably:• We answer a question of Gyárfás et al. whether there are n-vertex hypergraphs with irregularity strength greater than 2n. In fact we show that there are n-vertex hypergraphs with irregularity strength at least n2−o(1).• In addition, our results imply that s∗(n)=n2/w(n) where s∗(n) is the distinguishing closed-neighborhood number, i.e., the smallest integer N such that any n-vertex graph allows for a vertex labeling with positive integers at most N so that the sums of labels on distinct closed neighborhoods of vertices are distinct.

Constructing sparsest [formula omitted]-hamiltonian saturated [formula omitted]-uniform hypergraphs for a wide range of [formula omitted

Given k≥3 and 1≤ℓ<k, an (ℓ,k)-cycle is one in which consecutive edges, each of size k, overlap in exactly ℓ vertices. We study the smallest number of edges in k-uniform n-vertex hypergraphs which do not contain hamiltonian (ℓ,k)-cycles, but once a new edge is added, such a cycle is promptly created. It has been conjectured that this number is of order nℓ and confirmed for ℓ∈{1,k/2,k−1}, as well as for the upper range 0.8k≤ℓ≤k−1. Here we extend the validity of this conjecture to the lower–middle range (k−1)/3≤ℓ<(k−1)/2.

A note on minimum degree condition for Hamilton (a,b)-cycles in hypergraphs

Let k,a,b be positive integers with a+b=k. A k-uniform hypergraph is called an (a,b)-cycle if there is a partition (A0,B0,A1,B1,…,At−1,Bt−1) of the vertex set with |Ai|=a, |Bi|=b such that Ai∪Bi and Bi∪Ai+1 (subscripts module t) are edges for all i=0,1,…,t−1. Let H be a k-uniform n-vertex hypergraph with n≥5k and n divisible by k. By applying the concentration inequality for intersections of a uniform hypergraph with a random matching developed by Frankl and Kupavskii, we show that if there exists α∈(0,1) such that δa(H)≥(α+o(1))(n−ab) and δb(H)≥(1−α+o(1))(n−ba), then H contains a Hamilton (a,b)-cycle. As a corollary, we prove that if δℓ(H)≥(1/2+o(1))(n−ℓk−ℓ) for some ℓ≥k/2, then H contains a Hamilton (k−ℓ,ℓ)-cycle and this is asymptotically best possible.

Extremal problems in uniformly dense hypergraphs

For a k-uniform hypergraph F let ex(n,F) be the maximum number of edges of a k-uniform n-vertex hypergraph H which contains no copy of F. Determining or estimating ex(n,F) is a classical and central problem in extremal combinatorics. While for graphs (k=2) this problem is well understood, due to the work of Mantel, Turán, Erdős, Stone, Simonovits and many others, only very little is known for k-uniform hypergraphs for k>2. Already the case when F is a k-uniform hypergraph with three edges on k+1 vertices is still wide open even for k=3.We consider variants of such problems where the large hypergraph H enjoys additional hereditary density conditions. Questions of this type were suggested by Erdős and Sós about 30 years ago. In recent work with Rödl and Schacht it turned out that the regularity method for hypergraphs, established by Gowers and by Rödl et al. about a decade ago, is a suitable tool for extremal problems of this type and we shall discuss some of those recent results and some interesting open problems in this area.

On [formula omitted]-uniform hypergraphs with circumference less than [formula omitted

We show that for each k≥4 and n>r≥k+1, every n-vertex hypergraph with edge sizes at least r and no Berge cycle of length at least k has at most (k−1)(n−1)r edges. The bound is exact, and we describe the extremal hypergraphs. This implies and slightly refines the theorem of Győri, Katona and Lemons that for n>r≥k≥3, every n-vertex r-uniform hypergraph with no Berge path of length k has at most (k−1)nr+1 edges. To obtain the bounds, we study bipartite graphs with no cycles of length at least 2k, and then translate the results into the language of multi-hypergraphs.

A note on Ramsey numbers for Berge-[formula omitted] hypergraphs

For a graph G=(V,E), a hypergraph H is called Berge-G if there is a bijection ϕ:E(G)→E(H) such that for each e∈E(G), e⊆ϕ(e). The set of all Berge-G hypergraphs is denoted B(G).For integers k≥2, r≥2, and a graph G, let the Ramsey number Rr(B(G),k) be the smallest integer n such that no matter how the edges of a complete r-uniform n-vertex hypergraph are colored with k colors, there is a copy of a monochromatic Berge-G subhypergraph. Furthermore, let R(B(G),k) be the smallest integer n such that no matter how all subsets of an n-element set are colored with k colors, there is a monochromatic copy of a Berge-G hypergraph.We give an upper bound for Rr(B(G),k) in terms of graph Ramsey numbers. In particular, we prove that when G becomes acyclic after removing some vertex, Rr(B(G),k)≤4k|V(G)|+r−2, in contrast with classical multicolor Ramsey numbers.When G is a triangle (or a K4), we find sharper bounds and some exact results and determine some “small” Ramsey numbers: •k∕2−o(k)≤R3(B(K3),k)≤3k∕4+o(k),•For any odd integer t≠3, R(B(K3),2t−1)=t+2,•2ck≤R3(B(K4),k)≤e(1+o(1))(k−1)k!,•R3(B(K3),2)=R3(B(K3),3)=5,R3(B(K3),4)=6,R3(B(K3),5)=7,R3(B(K3),6)=8,R3(B(K3),8)=9,R3(B(K4),2)=6.

On the maximum size of connected hypergraphs without a path of given length

In this note we asymptotically determine the maximum number of hyperedges possible in an r-uniform, connected n-vertex hypergraph without a Berge path of length k, as n and k tend to infinity. We show that, unlike in the graph case, the multiplicative constant is smaller with the assumption of connectivity.

Stability of extremal hypergraphs with applications to an edge-coloring problem

We obtain sharper stability results for hypergraphs that avoid a copy of an expanded complete 2-graph and for hypergraphs that avoid a Fan subhypergraph. We apply this to an edge-coloring problem on uniform hypergraphs, where we wish to find an n-vertex hypergraph with the largest number of colorings avoiding a rainbow copy of a fixed hypergraph F.

Sparsification Upper and Lower Bounds for Graph Problems and Not-All-Equal SAT

We present several sparsification lower and upper bounds for classic problems in graph theory and logic. For the problems 4-Coloring, (Directed) Hamiltonian Cycle, and (Connected) Dominating Set, we prove that there is no polynomial-time algorithm that reduces any n-vertex input to an equivalent instance, of an arbitrary problem, with bitsize O(n^{2-varepsilon }) for varepsilon > 0, unless mathsf {NP subseteq coNP/poly} and the polynomial-time hierarchy collapses. These results imply that existing linear-vertex kernels for k-Nonblocker and k-Max Leaf Spanning Tree (the parametric duals of (Connected) Dominating Set) cannot be improved to have O(k^{2-varepsilon }) edges, unless mathsf {NP subseteq coNP/poly}. We also present a positive result and exhibit a non-trivial sparsification algorithm for d-Not-All-Equal-SAT. We give an algorithm that reduces an n-variable input with clauses of size at most d to an equivalent input with O(n^{d-1}) clauses, for any fixed d. Our algorithm is based on a linear-algebraic proof of Lovász that bounds the number of hyperedges in critically 3-chromatic d-uniform n-vertex hypergraphs by left( {begin{array}{c}n d-1end{array}}right) . We show that our kernel is tight under the assumption that mathsf {NP} nsubseteq mathsf {coNP}/mathsf {poly}.

Edge-colorings of uniform hypergraphs avoiding monochromatic matchings

For positive integers k, ℓ and r, and an r-uniform hypergraph H, let ck,ℓ,r(H) be the number of k-colorings of the set of hyperedges of H with no ℓ independent hyperedges of the same color. Let Hn,r denote the set of all n-vertex r-uniform hypergraphs and consider the function ck,ℓ,r(n)=max {ck,ℓ,r(H):H∈Hn,r}, the maximum of ck,ℓ,r(H) over all r-uniform hypergraphs H on n vertices. In this paper, we determine, for all fixed values of r, k and ℓ, and large n, the r-uniform n-vertex hypergraphs H with ck,ℓ,r(n)=ck,ℓ,r(H).

Coloring [Formula: see text]-Embeddable [Formula: see text]-Uniform Hypergraphs.

This paper extends the scenario of the Four Color Theorem in the following way. Let fancyscript{H}_{d,k} be the set of all k-uniform hypergraphs that can be (linearly) embedded into mathbb {R}^d. We investigate lower and upper bounds on the maximum (weak) chromatic number of hypergraphs in fancyscript{H}_{d,k}. For example, we can prove that for dge 3 there are hypergraphs in fancyscript{H}_{2d-3,d} on n vertices whose chromatic number is Omega (log n/log log n), whereas the chromatic number for n-vertex hypergraphs in fancyscript{H}_{d,d} is bounded by {mathcal {O}}(n^{(d-2)/(d-1)}) for dge 3.

Maximum hypergraphs without regular subgraphs

We show that an n-vertex hypergraph with no r-regular subgraphs has at most 2n−1 + r− 2 edges. We conjecture that if n > r, then every n-vertex hypergraph with no r-regular subgraphs having the maximum number of edges contains a full star, that is, 2n−1 distinct edges containing a given vertex. We prove this conjecture for n ≥ 425. The condition that n > r cannot be weakened.

Linear trees in uniform hypergraphs

Given a tree T on v vertices and an integer k≥2 one can define the k-expansion T(k) as a k-uniform linear hypergraph by enlarging each edge with a new, distinct set of k−2 vertices. T(k) has v+(v−1)(k−2) vertices. The aim of this paper is to show that using the delta-system method one can easily determine asymptotically the size of the largest T(k)-free n-vertex hypergraph, i.e., the Turán number of T(k).

On the Maximum Number of Edges in a Triple System Not Containing a Disjoint Family of a Given Size

In 1965 Erdős conjectured a formula for the maximum number of edges in a k-uniform n-vertex hypergraph without a matching of size s. We prove this conjecture for k = 3 and all s ≥ 1 and n ≥ 4s.

Dirac-type results for loose Hamilton cycles in uniform hypergraphs

A classic result of G.A. Dirac in graph theory asserts that for n ⩾ 3 every n-vertex graph with minimum degree at least n / 2 contains a spanning (so-called Hamilton) cycle. G.Y. Katona and H.A. Kierstead suggested a possible extension of this result for k-uniform hypergraphs. There a Hamilton cycle of an n-vertex hypergraph corresponds to an ordering of the vertices such that every k consecutive (modulo n) vertices in the ordering form an edge. Moreover, the minimum degree is the minimum ( k − 1 ) -degree, i.e. the minimum number of edges containing a fixed set of k − 1 vertices. V. Rödl, A. Ruciński, and E. Szemerédi verified (approximately) the conjecture of Katona and Kierstead and showed that every n-vertex, k-uniform hypergraph with minimum ( k − 1 ) -degree ( 1 / 2 + o ( 1 ) ) n contains such a tight Hamilton cycle. We study the similar question for Hamilton ℓ-cycles. A Hamilton ℓ-cycle in an n-vertex, k-uniform hypergraph ( 1 ⩽ ℓ < k ) is an ordering of the vertices and an ordered subset of the edges such that each such edge corresponds to k consecutive (modulo n) vertices and two consecutive edges intersect in precisely ℓ vertices. We prove sufficient minimum ( k − 1 ) -degree conditions for Hamilton ℓ-cycles if ℓ < k / 2 . In particular, we show that for every ℓ < k / 2 every n-vertex, k-uniform hypergraph with minimum ( k − 1 ) -degree ( 1 / ( 2 ( k − ℓ ) ) + o ( 1 ) ) n contains such a loose Hamilton ℓ-cycle. This degree condition is approximately tight and was conjectured by D. Kühn and D. Osthus (for ℓ = 1 ), who verified it when k = 3 . Our proof is based on the so-called weak regularity lemma for hypergraphs and follows the approach of V. Rödl, A. Ruciński, and E. Szemerédi.

Extremal problems for ordered hypergraphs: small patterns and some enumeration

Functions ex e ( F, n) and ex i ( F, n) count edges and vertex-edge incidences in extremal n-vertex hypergraphs avoiding, in the ordered sense, a hypergraph F. We determine them exactly for F consisting of distinct singleton edges and give a survey of exact results for the 55 hypergraphs F with at most four incidences. In the enumerative part we give formulae and recurrences for the numbers of hypergraphs with n incidences. We find rough asymptotics of these numbers and derive identities analogous to Dobiǹski's formula for Bell numbers.

Hypergraphs, the Qualitative Solvability of κ · λ = 0, and Volterra Multipliers for Nonlinear Dynamical Systems

Certain sign equivalence classes of n-dimensional nonlinear dynamical systems correspond to n-vertex hypergraphs, The global stability of some such dynamical systems can be guaranteed if the associated hypergraphs have a simplicity of structure and meet certain quantitative path product conditions. A purely algebraic version of the same problem can be described as follows. Suppose we are given a rectangular matrix pattern of signs; each entry in the matrix is +, −, or 0. For every real matrix κ of the same sign pattern, is there a real vector λ, each component of which is positive, such that κ · λ = 0? This paper presents graph theoretic sufficient conditions on a hypergraph generated from the sign pattern of κ which guarantee the existence of λ. For κ with more highly connected hypergraphs, this paper also presents sufficient qualitative conditions on the sign pattern of κ and certain quantitative conditions on sums of hypergraph path products which together guarantee the existence of λ.