K-uniform Hypergraph Research Articles

Uniform hypergraphs with the first two smallest spectral radii

The spectral radius of a uniform hypergraph G is the maximum modulus of the eigenvalues of the adjacency tensor of G. For k≥2, among connected k-uniform hypergraphs with m≥1 edges, we show that the k-uniform loose path with m edges is the unique one with smallest spectral radius, and we also determine the unique ones with second smallest spectral radius when m≥2.

Hamiltonicity in randomly perturbed hypergraphs

For integers k≥3 and 1≤ℓ≤k−1, we prove that for any α>0, there exist ϵ>0 and C>0 such that for sufficiently large n∈(k−ℓ)N, the union of a k-uniform hypergraph with minimum vertex degree αnk−1 and a binomial random k-uniform hypergraph G(k)(n,p) with p≥n−(k−ℓ)−ϵ for ℓ≥2 and p≥Cn−(k−1) for ℓ=1 on the same vertex set contains a Hamiltonian ℓ-cycle with high probability. Our result is best possible up to the values of ϵ and C and answers a question of Krivelevich, Kwan and Sudakov.

On some properties of the α-spectral radius of the k-uniform hypergraph

In this paper we show how the α-spectral radius changes under the edge grafting operations on connected k-uniform hypergraphs. We characterize the extremal hypertree for α-spectral radius among k-uniform non-caterpillar hypergraphs with given order, size and diameter. We also characterize the supertree with the second largest α-spectral radius among all k-uniform supertrees on n vertices.

Estimating the [formula omitted]-colorability threshold for a random hypergraph

The paper deals with estimating the r-colorability threshold for a random k-uniform hypergraph in the binomial model H(n,k,p). We consider the sparse case, when the expected number of edges is a linear function of n, pnk=cn, and c>0, k⩾4, r⩾3 are some fixed constants. We show that if c<rk−1lnr−12lnr−r−1r+O(k2r1−k∕3lnr) then H(n,k,cn∕nk) is r-colorable with probability tending to 1 as n→∞.

Complete colourings of hypergraphs

A complete c-colouring of a graph is a proper colouring in which every pair of distinct colours from [c]={1,2,…,c} appears as the colours of endvertices of some edge. We consider the following generalisation of this concept to uniform hypergraphs. A complete c-colouring for a k-uniform hypergraph H is a mapping from the vertex set of H to [c], such that (i) the colour set used on each edge has exactly k elements, and (ii) every k-element subset of [c] appears as the colour set of some edge. In this paper we exhibit some differences between complete colourings of graphs and hypergraphs. First, it is known that every graph has a complete c-colouring for some c.In contrast, we show an infinite family of hypergraphs H that do not admit a complete c-colouring for any c. We also extend this construction to λ-complete colourings (for 0<λ≤1), where condition (ii) is substituted with: at least λck different colour sets appear on edges.We establish upper and lower bounds on a maximum degree, which guarantees the existence of a complete colouring of any hypergraph. Moreover, we prove that it is NP-complete to determine if a given hypergraph has a λ-complete colouring. Next, we show that, unlike graphs, hypergraphs do not have the so-called interpolation property, i.e., we construct hypergraphs that have a complete r-colouring and a complete s-colouring, but no complete t-colouring for some t such that r<t<s.Finally, we investigate the notion of λ-complete colourings of graphs (i.e., 2-uniform hypergraphs). We show that λ-complete colourings have the same interpolation property as complete colourings. Moreover, we prove that it is NP-complete to decide whether a tree with c2 edges has a λ-complete c-colouring, which strengthens the result by Cairnie and Edwards (1997).

A Tight Bound for Hyperaph Regularity

The hypergraph regularity lemma—the extension of Szemeredi’s graph regularity lemma to the setting of k-uniform hypergraphs—is one of the most celebrated combinatorial results obtained in the past decade. By now there are several (very different) proofs of this lemma, obtained by Gowers, by Nagle–Rodl–Schacht–Skokan and by Tao. Unfortunately, what all these proofs have in common is that they yield regular partitions whose order is given by the k-th Ackermann function. We show that such Ackermann-type bounds are unavoidable for every $$k \ge 2$$ , thus confirming a prediction of Tao. Prior to our work, the only result of this type was Gowers’ famous lower bound for graph regularity.

Hypergraph cuts above the average

An r-cut of a k-uniform hypergraph H is a partition of the vertex set of H into r parts and the size of the cut is the number of edges which have a vertex in each part. A classical result of Edwards says that every m-edge graph has a 2-cut of size $$m{\rm/}2 + {\rm\Omega)}(\sqrt m)$$ and this is best possible. That is, there exist cuts which exceed the expected size of a random cut by some multiple of the standard deviation. We study analogues of this and related results in hypergraphs. First, we observe that similarly to graphs, every m-edge k-uniform hypergraph has an r-cut whose size is $${\rm\Omega}(\sqrt m)$$ larger than the expected size of a random r-cut. Moreover, in the case where k = 3 and r = 2 this bound is best possible and is attained by Steiner triple systems. Surprisingly, for all other cases (that is, if k ≥ 4 or r ≥ 3), we show that every m-edge k-uniform hypergraph has an r-cut whose size is Ω(m5/9) larger than the expected size of a random r-cut. This is a significant difference in behaviour, since the amount by which the size of the largest cut exceeds the expected size of a random cut is now considerably larger than the standard deviation.

Codegree Conditions for Tiling Complete k-Partite k-Graphs and Loose Cycles

AbstractGiven twok-graphs (k-uniform hypergraphs)FandH, a perfectF-tiling (orF-factor) inHis a set of vertex-disjoint copies ofFthat together cover the vertex set ofH. For all completek-partitek-graphsK, Mycroft proved a minimum codegree condition that guarantees aK-factor in ann-vertexk-graph, which is tight up to an error termo(n). In this paper we improve the error term in Mycroft’s result to a sublinear term that relates to the Turán number ofKwhen the differences of the sizes of the vertex classes ofKare co-prime. Furthermore, we find a construction which shows that our improved codegree condition is asymptotically tight in infinitely many cases, thus disproving a conjecture of Mycroft. Finally, we determine exact minimum codegree conditions for tilingK(k)(1, … , 1, 2) and tiling loose cycles, thus generalizing the results of Czygrinow, DeBiasio and Nagle, and of Czygrinow, respectively.

Signed k-uniform hypergraphs and tensors

A signed k-graph is a k-graph with an incidence function on each vertex and its incident edge. In this paper we extend some concepts of the adjacency and Laplacian tensor from k-graphs to signed k-graphs. Based on the spectral properties of the real symmetric tensor, we investigate the H-spectrum of adjacency and Laplacian tensors for signed k-graphs. Finally the inverse Perron value of any vertex is present. Some bounds for the inverse Perron value are established in terms of some invariants including the number of vertices (edges), the degree sequence.

A note on the minimum number of edges in hypergraphs with property O

An oriented k-graph is said to have Property O if for every linear order of the vertex set, there is some edge oriented consistently with the linear order. Recently Duffus, Kay and Rödl investigated the minimum number f(k) of edges in a k-uniform hypergraph with Property O. They proved that k!≤f(k)≤(k2lnk)k!, where the upper bound holds for sufficiently large k. In this short note we improve their upper bound by a factor of klnk showing that f(k)≤⌊k2⌋+1k!−⌊k2⌋(k−1)! for every k≥3. We also show that their lower bound is not tight. Furthermore, Duffus, Kay and Rödl also studied the minimum possible number n(k) of vertices in an oriented k-graph with Property O. For k=3 they showed that n(3)∈{6,7,8,9}, and asked for the precise value of n(3). Here we show that n(3)=6.

Saturation of Berge hypergraphs

Given a graph F, a hypergraph is a Berge- F if it can be obtained by expanding each edge in F to a hyperedge containing it. A hypergraph H is Berge-F-saturated if H does not contain a subhypergraph that is a Berge-F, but for any edge e∈E(H¯), H+e does. The k-uniform saturation number of Berge-F is the minimum number of edges in a k-uniform Berge-F-saturated hypergraph on n vertices. For k=2 this definition coincides with the classical definition of saturation for graphs. In this paper we study the saturation numbers for Berge triangles, paths, cycles, stars and matchings in k-uniform hypergraphs.

Linearity of saturation for Berge hypergraphs

For a graph F, we say a hypergraph H is a Berge-F if it can be obtained from F by replacing each edge of F with a hyperedge containing it. We say a hypergraph is Berge-F-saturated if it does not contain a Berge-F, but adding any hyperedge creates a copy of a Berge- F. The k-uniform saturation number of Berge-F, satk(n,Berge-F) is the fewest number of hyperedges in a Berge-F-saturated k-uniform hypergraph on n vertices. We show that satk(n,Berge-F)=O(n) for all graphs F and uniformities 3≤k≤5, partially answering a conjecture of English, Gordon, Graber, Methuku, and Sullivan. We also extend this conjecture to Berge copies of hypergraphs.

Panchromatic 3-colorings of random hypergraphs

The paper deals with panchromatic 3-colorings of random hypergraphs. A vertex 3-coloring is said to be panchromatic for a hypergraph if every color can be found on every edge. Let H(n,k,p) denote the binomial model of a random k-uniform hypergraph on n vertices. For given fixed c>0, k⩾3 and p=cn∕nk, we prove that if c<ln33⋅32k−ln32−O32kthen H(n,k,p) admits a panchromatic 3-coloring with probability tending to 1 as n→∞, but if k is large enough and c>ln33⋅32k−ln32+O34kthen H(n,k,p) does not admit a panchromatic 3-coloring with probability tending to 1 as n→∞.

Upper bounds for signless Laplacian Z-spectral radius of uniform hypergraphs

Let ℋ be a k-uniform hypergraph on n vertices with degree sequence Δ = d1 ⩾ ⋯ ⩾ dn = δ In this paper, in terms of degree di; we give some upper bounds for the Z-spectral radius of the signless Laplacian tensor (Q(ℋ)) of ℋ. Some examples are given to show the efficiency of these bounds.

Almost t-complementary uniform hypergraphs

An almost t-complementary k-hypergraph is a k-uniform hypergraph with vertex set V and edge set E for which there exists a permutation $$\theta \in Sym(V)$$ such that the sets $$E, E^\theta , E^{\theta ^2}, \ldots , E^{\theta ^{t-1}}$$ partition the set of all k-subsets of V minus one edge. Such a permutation $$\theta $$ is called an almost (t, k)-complementing permutation. Almost t-complementary k-hypergraphs are a natural generalization of almost self-complementary graphs, which were previously studied by Clapham, Kamble et al., and Wojda. We prove that there exists an almost p-complementary k-hypergraph of order n whenever the base-p representation of k is a subsequence of the base-p representation of n, where p is prime.

The (signless Laplacian) spectral radius (of subgraphs) of uniform hypergraphs

Let ?1(G) and q1(G) be the spectral radius and the signless Laplacian spectral radius of a kuniform hypergraph G, respectively. In this paper, we give the lower bounds of d-?1(H) and 2d-q1(H), where H is a proper subgraph of a f (-edge)-connected d-regular (linear) k-uniform hypergraph. Meanwhile, we also give the lower bounds of 2?-q1(G) and ?-?1(G), where G is a nonregular f (-edge)-connected (linear) k-uniform hypergraph with maximum degree ?.

Domination game on uniform hypergraphs

In this paper we introduce and study the domination game on hypergraphs. This is played on a hypergraph H by two players, namely Dominator and Staller, who alternately select vertices such that each selected vertex enlarges the set of vertices dominated so far. The game is over if all vertices of H are dominated. Dominator aims to finish the game as soon as possible, while Staller aims to delay the end of the game. If each player plays optimally and Dominator starts, the length of the game is the invariant ‘game domination number’ denoted by γg(H). This definition is the generalization of the domination game played on graphs and it is a special case of the transversal game on hypergraphs. After some basic general results, we establish an asymptotically tight upper bound on the game domination number of k-uniform hypergraphs. In the remaining part of the paper we prove that γg(H)≤5n∕9 if H is a 3-uniform hypergraph of order n and does not contain isolated vertices. This also implies the following new result for graphs: If G is an isolate-free graph on n vertices and each of its edges is contained in a triangle, then γg(G)≤5n∕9.

Erdős–Hajnal Conjecture for Graphs with Bounded VC-Dimension

The Vapnik–Chervonenkis dimension (in short, VC-dimension) of a graph is defined as the VC-dimension of the set system induced by the neighborhoods of its vertices. We show that every n-vertex graph with bounded VC-dimension contains a clique or an independent set of size at least $$e^{(\log n)^{1 - o(1)}}$$ . The dependence on the VC-dimension is hidden in the o(1) term. This improves the general lower bound, $$e^{c\sqrt{\log n}}$$ , due to Erdős and Hajnal, which is valid in the class of graphs satisfying any fixed nontrivial hereditary property. Our result is almost optimal and nearly matches the celebrated Erdős–Hajnal conjecture, according to which one can always find a clique or an independent set of size at least $$e^{\Omega (\log n)}$$ . Our results partially explain why most geometric intersection graphs arising in discrete and computational geometry have exceptionally favorable Ramsey-type properties. Our main tool is a partitioning result found by Lovasz–Szegedy and Alon–Fischer–Newman, which is called the “ultra-strong regularity lemma” for graphs with bounded VC-dimension. We extend this lemma to k-uniform hypergraphs, and prove that the number of parts in the partition can be taken to be $$(1/\varepsilon )^{O(d)}$$ , improving the original bound of $$(1/\varepsilon )^{O(d^2)}$$ in the graph setting. We show that this bound is tight up to an absolute constant factor in the exponent. Moreover, we give an $$O(n^k)$$ -time algorithm for finding a partition meeting the requirements. Finally, we establish tight bounds on Ramsey–Turan numbers for graphs with bounded VC-dimension.

Large monochromatic components and long monochromatic cycles in random hypergraphs

We extend results of Gyárfás and Füredi on the largest monochromatic component in r-colored complete k-uniform hypergraphs to the setting of random hypergraphs. We also study long monochromatic loose cycles in r-colored random hypergraphs. In particular, we obtain a random analog of a result of Gyárfás, Sárközy, and Szemerédi on the longest monochromatic loose cycle in 2-colored complete k-uniform hypergraphs.

Some Ore-type Results for Matching and Perfect Matching in k-uniform Hypergraphs

Let S1 and S2 be two (k − 1)-subsets in a k-uniform hypergraph H. We call S1 and S2 strongly or middle or weakly independent if H does not contain an edge e ∈ E(H) such that S1 ∩ e ≠ ∅ and S2 ∩ e ≠ ∅ or e ⊆ S1 ∪ S2 or e ⊇ S1 ∪ S2, respectively. In this paper, we obtain the following results concerning these three independence. (1) For any n ≥ 2k2 − k and k ≥ 3, there exists an n-vertex k-uniform hypergraph, which has degree sum of any two strongly independent (k − 1)-sets equal to 2n−4(k−1), contains no perfect matching; (2) Let d ≥ 1 be an integer and H be a k-uniform hypergraph of order n ≥ kd+(k−2)k. If the degree sum of any two middle independent (k−1)-subsets is larger than 2(d−1), then H contains a d-matching; (3) For all k ≥ 3 and sufficiently large n divisible by k, we completely determine the minimum degree sum of two weakly independent (k − 1)-subsets that ensures a perfect matching in a k-uniform hypergraph H of order n.