R-uniform Hypergraph Research Articles

Exponential Lower Bound for Berge-Ramsey Problems

We give an exponential lower bound for the smallest N such that no matter how we c-color the edges of a complete r-uniform hypergraph on N vertices, we can always find a monochromatic Berge-K_n.

Decomposing uniform hypergraphs into uniform hypertrees and single edges

Given two r-uniform hypergraphs G and H, an H-decomposition of G is a partition of the edge set of G such that each part is either a single edge or forms a hypergraph isomorphic to H. Let ϕr(n,H) be the smallest integer such that any r-uniform hypergraph G of order n admits an H-decomposition with at most ϕr(n,H) parts. In this paper we determine the exact value of ϕr(n,H) when H is an arbitrary r-uniform hypertree with t edges.

Hyperstar Decomposition of r-partite complete, Knodel and Fibonacci Hypergraphs

A graph is a representation of a relation between a pair of objects in a set called vertex set. A relation is a set of two-element subsets of the vertex set, called edge set. If the restriction of the two-element subset is relaxed to any number, the corresponding representation is a hypergraph. An r-uniform hypergraph is a Steiner system, in which every pair of nodes is contained in exactly a single edge. Knodel graphs, Fibonacci graphs, and r-partite complete graphs are structures with optimal communication properties. Knodel hypergraph and Fibonacci hypergraph are introduced. Partitioning of the edge set leads to the decomposition of a given graph into sub-structures. If all such sub-structures are isomorphic, the decomposition is an isomorphic decomposition. The isomorphic decomposition of an r-partite hypergraph, Knodel hypergraph, and Fibonacci hypergraph into hyperstars are presented.

Lagrangians of hypergraphs II: When colex is best

A well-known conjecture of Frankl and Füredi from 1989 states that an initial segment of the colexicographic order has the largest Lagrangian of any r-uniform hypergraph with m hyperedges. We show that this is true when r = 3. We also give a new proof of a related conjecture of Nikiforov for large t and a counterexample to an old conjecture of Ahlswede and Katona.

Multicovering hypergraphs

Let fr(n,p) represent the minimum number of complete r-partite r-graphs required to cover every edge of the complete r-uniform hypergraph on n vertices at least once and at most p times.Graham–Pollak theorem states that f2(n,1)=n−1. Upper and lower bounds were known for r=2 and a general p. In this note we give bounds for fr(n,p) for general r and p.

Sierpiński products of r-uniform hypergraphs

If H1 and H2 are r-uniform hypergraphs and f is a function from the set of all (r − 1)-element subsets of V(H1) into V(H2), then the Sierpinski product H1⊗fH2 is defined to have vertex set V(H1) × V(H2) and hyperedges falling into two classes: (g, h1)(g, h2)⋯(g, hr), such that g ∈ V(H1) and h1h2⋯hr ∈ E(H2),and (g1, f({g2, g3, …, gr}))(g2, f({g1, g3, …, gr}))⋯(gr, f({g1, g2, …, gr − 1})),such that g1g2⋯gr ∈ E(H1). We develop the basic structure possessed by this product and offer proofs of numerous extremal properties involving connectivity, clique numbers, and strong chromatic numbers.

A proof of a conjecture of Gyárfás, Lehel, Sárközy and Schelp on Berge-cycles

AbstractIt has been conjectured that, for any fixed \[{\text{r}} \geqslant 2\] and sufficiently large n, there is a monochromatic Hamiltonian Berge-cycle in every \[({\text{r}} - 1)\]-colouring of the edges of \[{\text{K}}_{\text{n}}^{\text{r}}\], the complete r-uniform hypergraph on n vertices. In this paper we prove this conjecture.

Lagrangian densities of short 3-uniform linear paths and Turán numbers of their extensions

For a fixed positive integer n and an r-uniform hypergraph H, the Turan number ex(n, H) is the maximum number of edges in an H-free r-uniform hypergraph on n vertices, and the Lagrangian density of H is defined as $$\pi _{\lambda }(H)=\sup \{r! \lambda (G) : G \;\text {is an}\; H\text {-free} \;r\text {-uniform hypergraph}\}$$ , where $$\lambda (G)=\max \{\sum _{e \in G}\prod \limits _{i\in e}x_{i}: x_i\ge 0 \;\text {and}\; \sum _{i\in V(G)} x_i=1\}$$ is the Lagrangian of G. For an r-uniform hypergraph H on t vertices, it is clear that $$\pi _{\lambda }(H)\ge r!\lambda {(K_{t-1}^r)}$$ . Let us say that an r-uniform hypergraph H on t vertices is $$\lambda $$ -perfect if $$\pi _{\lambda }(H)= r!\lambda {(K_{t-1}^r)}$$ . A result of Motzkin and Straus implies that all graphs are $$\lambda $$ -perfect. It is interesting to explore what kind of hypergraphs are $$\lambda $$ -perfect. A conjecture in [22] proposes that every sufficiently large r-uniform linear hypergraph is $$\lambda $$ -perfect. In this paper, we investigate whether the conjecture holds for linear 3-uniform paths. Let $$P_t=\{e_1, e_2, \dots , e_t\}$$ be the linear 3-uniform path of length t, that is, $$|e_i|=3$$ , $$|e_i \cap e_{i+1}|=1$$ and $$e_i \cap e_j=\emptyset $$ if $$|i-j|\ge 2$$ . We show that $$P_3$$ and $$P_4$$ are $$\lambda $$ -perfect. Applying the results on Lagrangian densities, we determine the Turan numbers of their extensions.

Counting independent sets in regular hypergraphs

Amongst d-regular r-uniform hypergraphs on n vertices, which ones have the largest number of independent sets? While the analogous problem for graphs (originally raised by Granville) is now well-understood, it is not even clear what the correct general conjecture ought to be; our goal here is to propose such a generalisation. Lending credence to our conjecture, we verify it within the class of ‘quasi-bipartite’ hypergraphs (a generalisation of bipartite graphs that seems natural in this context) by adopting the entropic approach of Kahn.

A sharp threshold for bootstrap percolation in a random hypergraph

Given a hypergraph H, the H-bootstrap process starts with an initial set of infected vertices of H and, at each step, a healthy vertex v becomes infected if there exists a hyperedge of H in which v is the unique healthy vertex. We say that the set of initially infected vertices percolates if every vertex of H is eventually infected. We show that this process exhibits a sharp threshold when H is a hypergraph obtained by randomly sampling hyperedges from an approximately d-regular r-uniform hypergraph satisfying some mild degree and codegree conditions; this confirms a conjecture of Morris. As a corollary, we obtain a sharp threshold for a variant of the graph bootstrap process for strictly 2-balanced graphs which generalises a result of Korándi, Peled and Sudakov. Our approach involves an application of the differential equations method.

The feasible region of hypergraphs

Let F be a family of r-uniform hypergraphs. The feasible region Ω(F) of F is the set of points (x,y) in the unit square such that there exists a sequence of F-free r-uniform hypergraphs whose shadow density approaches x and whose edge density approaches y. The feasible region provides a lot of combinatorial information, for example, the supremum of y over all (x,y)∈Ω(F) is the Turán density π(F), and Ω(∅) gives the Kruskal-Katona theorem.We undertake a systematic study of Ω(F), and prove that Ω(F) is completely determined by a left-continuous almost everywhere differentiable function; and moreover, there exists an F for which this function is not continuous. We also extend some old related theorems. For example, we generalize a result of Fisher and Ryan to hypergraphs and extend a classical result of Bollobás by almost completely determining the feasible region for cancellative triple systems.

A linear hypergraph extension of the bipartite Turán problem

An r-uniform hypergraph is linear if every two edges intersect in at most one vertex. Given a family of r-uniform hypergraphs F, the linear Turán number exrlin(n,F) is the maximum number of edges of a linear r-uniform hypergraph on n vertices that does not contain any member of F as a subhypergraph.Given a graph F and a positive integer r≥2, the r-expansion of F is the r-graph F+ obtained from F by enlarging each edge of F with r−2 new vertices disjoint from V(F) such that distinct edges of F are enlarged by distinct vertices. For t≥s≥2, we prove the following extension of Kővári–Sós–Turán’s theorem exrlin(n,Ks,t+)≤(t−1)1sr(r−1)⋅n2−1s+O(n2−2s). Specially, for s=2, r=3, we prove that ex3lin(n,K2,t+)=1+ot(1)16t−1⋅n3∕2, which is an improvement of Gerbner, Methuku and Vizer’s result (Gerbner et al., 2019). Moreover, we also prove some sharp bounds for the linear Turán number of Ks,t+.

Ryser's Conjecture for t-intersecting hypergraphs

A well-known conjecture, often attributed to Ryser, states that the cover number of an r-partite r-uniform hypergraph is at most r−1 times larger than its matching number. Despite considerable effort, particularly in the intersecting case, this conjecture remains wide open, motivating the pursuit of variants of the original conjecture. Recently, Bustamante and Stein and, independently, Király and Tóthmérész considered the problem under the assumption that the hypergraph is t-intersecting, conjecturing that the cover number τ(H) of such a hypergraph H is at most r−t. In these papers, it was proven that the conjecture is true for r≤4t−1, but also that it need not be sharp; when r=5 and t=2, one has τ(H)≤2.We extend these results in two directions. First, for all t≥2 and r≤3t−1, we prove a tight upper bound on the cover number of these hypergraphs, showing that they in fact satisfy τ(H)≤⌊(r−t)/2⌋+1. Second, we extend the range of t for which the conjecture is known to be true, showing that it holds for all r≤367t−5. We also introduce several related variations on this theme. As a consequence of our tight bounds, we resolve the problem for k-wise t-intersecting hypergraphs, for all k≥3 and t≥1. We further give bounds on the cover numbers of strictly t-intersecting hypergraphs and the s-cover numbers of t-intersecting hypergraphs.

Sparse hypergraphs: New bounds and constructions

Let fr(n,v,e) denote the maximum number of edges in an r-uniform hypergraph on n vertices, in which the union of any e distinct edges contains at least v+1 vertices. The study of fr(n,v,e) was initiated by Brown, Erdős and Sós more than forty years ago. In the literature, the following conjecture is well known.Conjecture:nk−o(1)<fr(n,er−(e−1)k+1,e)=o(nk) holds for all fixed integers r>k≥2 and e≥3 as n→∞.For r=3,e=3,k=2, the bound n2−o(1)<f3(n,6,3)=o(n2) was proved by the celebrated (6,3)-theorem of Ruzsa and Szemerédi. In this paper, we add more evidence for the validity of the conjecture. On one hand, using the hypergraph removal lemma we show that the upper bound part of the conjecture is true for all fixed integers r≥k+1≥e≥3. On the other hand, using tools from additive number theory we present several constructions showing that the lower bound part of the conjecture is true for r≥3, k=2 and e=4,5,7,8. Prior to our results, all known constructions that match the conjectured lower bound satisfy either r=3 or e=3. Our constructions are the first ones in the literature that break this barrier.

The extremal p-spectral radius of Berge hypergraphs

Let G be a graph. We say that a hypergraph H is a Berge-G if there is a bijection ϕ:E(G)→E(H) such that e⊆ϕ(e) for all e∈E(G). For any r-uniform hypergraph H and a real number p≥1, the p-spectral radius λ(p)(H) of H is defined asλ(p)(H):=maxx∈Rn,‖x‖p=1⁡r∑{i1,i2,…,ir}∈E(H)xi1xi2⋯xir. In this paper, we study the p-spectral radius of Berge-G hypergraphs. We determine the 3-uniform hypergraphs with maximum p-spectral radius for p≥1 among Berge-G hypergraphs when G is a path, a cycle or a star.

The average number of spanning hypertrees in sparse uniform hypergraphs

An r-uniform hypergraph H consists of a set of vertices V and a set of edges whose elements are r-subsets of V. We define a hypertree to be a connected hypergraph which contains no cycles. A hypertree spans a hypergraph H if it is a subhypergraph of H which contains all vertices of H. Greenhill et al. (2017) gave an asymptotic formula for the average number of spanning trees in graphs with given, sparse degree sequence. We prove an analogous result for r-uniform hypergraphs with given degree sequence k=(k1,…,kn). Our formula holds when r5kmax3=o((kr−k−r)n), where k is the average degree and kmax is the maximum degree.

Finding tight Hamilton cycles in random hypergraphs faster

AbstractIn an r-uniform hypergraph on n vertices, a tight Hamilton cycle consists of n edges such that there exists a cyclic ordering of the vertices where the edges correspond to consecutive segments of r vertices. We provide a first deterministic polynomial-time algorithm, which finds a.a.s. tight Hamilton cycles in random r-uniform hypergraphs with edge probability at least C log3n/n.Our result partially answers a question of Dudek and Frieze, who proved that tight Hamilton cycles exist already for p = ω(1/n) for r = 3 and p = (e + o(1))/n for $r \ge 4$ using a second moment argument. Moreover our algorithm is superior to previous results of Allen, Böttcher, Kohayakawa and Person, and Nenadov and Škorić, in various ways: the algorithm of Allen et al. is a randomized polynomial-time algorithm working for edge probabilities $p \ge {n^{ - 1 + \varepsilon}}$, while the algorithm of Nenadov and Škorić is a randomized quasipolynomial-time algorithm working for edge probabilities $p \ge C\mathop {\log }\nolimits^8 n/n$.

Hamiltonian Berge cycles in random hypergraphs

AbstractIn this note we study the emergence of Hamiltonian Berge cycles in random r-uniform hypergraphs. For $r\geq 3$ we prove an optimal stopping time result that if edges are sequentially added to an initially empty r-graph, then as soon as the minimum degree is at least 2, the hypergraph with high probability has such a cycle. In particular, this determines the threshold probability for Berge Hamiltonicity of the Erdős–Rényi random r-graph, and we also show that the 2-out random r-graph with high probability has such a cycle. We obtain similar results for weak Berge cycles as well, thus resolving a conjecture of Poole.

Partitioning ordered hypergraphs

An ordered r-graph is an r-uniform hypergraph whose vertex set is linearly ordered. Given 2≤k≤r, an ordered r-graph H is interval k-partite if there exist at least k disjoint intervals in the ordering such that every edge of H has nonempty intersection with each of the intervals and is contained in their union.Our main result implies that if α>k−1, then for each d>0 every n-vertex ordered r-graph with dnα edges has for some m≤n an m-vertex interval k-partite subgraph with Ω(dmα) edges. This is an extension to ordered r-graphs of the observation by Erdős and Kleitman that every r-graph contains an r-partite subgraph with a constant proportion of the edges. The restriction α>k−1 is sharp. We also present applications of the main result to several extremal problems for ordered hypergraphs.

Some tight lower bounds for Turán problems via constructions of multi-hypergraphs

Recently, several hypergraph Turán problems were solved by the powerful random algebraic method. However, the random algebraic method usually requires some parameters to be very large, hence we are concerned about how these Turán numbers depend on such large parameters of the forbidden hypergraphs. In this paper, we determine the dependence on such specified large constant for several hypergraph Turán problems. More specifically, for complete r-partite r-uniform hypergraphs, we show that if sr is sufficiently larger than s1,s2,…,sr−1, then exr(n,Ks1,s2,…,sr(r))=Θ(sr1s1s2⋯sr−1nr−1s1s2⋯sr−1).For complete bipartite r-uniform hypergraphs, we prove that if s is sufficiently larger than t, we have exr(n,Ks,t(r))=Θ(s1tnr−1t).In particular, our results imply that the famous Kővári–Sós–Turán’s upper bound ex(n,Ks,t)=O(t1sn2−1s) has the correct dependence on large t. The main approach is to construct random multi-hypergraph via a variant of random algebraic method.