scholarly journals The Maximum Number of Cliques in Hypergraphs without Large Matchings

10.37236/9604 ◽  
2020 ◽  
Vol 27 (4) ◽  
Author(s):  
Erica L.L. Liu ◽  
Jian Wang
Keyword(s):  
Sufficient Condition ◽  
Matching Number ◽  
Uniform Hypergraph ◽  
Rainbow Matching ◽  
Uniform Hypergraphs ◽  
Large N ◽  
Vertex Set

Let $[n]$ denote the set $\{1, 2, \ldots, n\}$ and $\mathcal{F}^{(r)}_{n,k,a}$ be an $r$-uniform hypergraph on the vertex set $[n]$ with edge set consisting of all the $r$-element subsets of $[n]$ that contains at least $a$ vertices in $[ak+a-1]$. For $n\geq 2rk$, Frankl proved that $\mathcal{F}^{(r)}_{n,k,1}$ maximizes the number of edges in $r$-uniform hypergraphs on $n$ vertices with the matching number at most $k$. Huang, Loh and Sudakov considered a multicolored version of the Erd\H{o}s matching conjecture, and provided a sufficient condition on the number of edges for a multicolored hypergraph to contain a rainbow matching of size $k$. In this paper, we show that $\mathcal{F}^{(r)}_{n,k,a}$ maximizes the number of $s$-cliques in $r$-uniform hypergraphs on $n$ vertices with the matching number at most $k$ for sufficiently large $n$, where $a=\lfloor \frac{s-r}{k} \rfloor+1$. We also obtain a condition on the number of $s$-clques for a multicolored $r$-uniform hypergraph to contain a rainbow matching of size $k$, which reduces to the condition of Huang, Loh and Sudakov when $s=r$.

scholarly journals Monochromatic Path and Cycle Partitions in Hypergraphs

10.37236/2631 ◽  
2013 ◽  
Vol 20 (1) ◽  
Author(s):  
András Gyárfás ◽  
Gábor N. Sárközy
Keyword(s):  
Uniform Hypergraph ◽  
Uniform Hypergraphs ◽  
Vertex Set

Here we address the problem to partition edge colored hypergraphs by monochromatic paths and cycles generalizing a well-known similar problem for graphs.We show that $r$-colored $r$-uniform complete hypergraphs can be partitioned into monochromatic Berge-paths of distinct colors. Also, apart from $2k-5$ vertices, $2$-colored $k$-uniform hypergraphs can be partitioned into two monochromatic loose paths.In general, we prove that in any $r$-coloring of a $k$-uniform hypergraph there is a partition of the vertex set intomonochromatic loose cycles such that their number depends only on $r$ and $k$.

scholarly journals The Bounds of the Edge Number in Generalized Hypertrees

Mathematics ◽  
2018 ◽  
Vol 7 (1) ◽  
pp. 2 ◽  
Author(s):  
Ke Zhang ◽  
Haixing Zhao ◽  
Zhonglin Ye ◽  
Yu Zhu ◽  
Liang Wei
Keyword(s):  
Upper Bounds ◽  
Uniform Hypergraph ◽  
Lower And Upper Bounds ◽  
Number Of Components ◽  
Uniform Hypergraphs ◽  
Vertex Set

A hypergraph H = ( V , ε ) is a pair consisting of a vertex set V , and a set ε of subsets (the hyperedges of H ) of V . A hypergraph H is r -uniform if all the hyperedges of H have the same cardinality r . Let H be an r -uniform hypergraph, we generalize the concept of trees for r -uniform hypergraphs. We say that an r -uniform hypergraph H is a generalized hypertree ( G H T ) if H is disconnected after removing any hyperedge E , and the number of components of G H T − E is a fixed value k   ( 2 ≤ k ≤ r ) . We focus on the case that G H T − E has exactly two components. An edge-minimal G H T is a G H T whose edge set is minimal with respect to inclusion. After considering these definitions, we show that an r -uniform G H T on n vertices has at least 2 n / ( r + 1 ) edges and it has at most n − r + 1 edges if r ≥ 3   and   n ≥ 3 , and the lower and upper bounds on the edge number are sharp. We then discuss the case that G H T − E has exactly k   ( 2 ≤ k ≤ r − 1 ) components.

scholarly journals The Minimum Number of Edges in Uniform Hypergraphs with Property O

2017 ◽  
Vol 27 (4) ◽  
pp. 531-538 ◽  
Author(s):  
DWIGHT DUFFUS ◽  
BILL KAY ◽  
VOJTĚCH RÖDL
Keyword(s):  
Linear Order ◽  
Uniform Hypergraph ◽  
Uniform Hypergraphs ◽  
Minimum Number ◽  
Vertex Set

An oriented k-uniform hypergraph (a family of ordered k-sets) has the ordering property (or Property O) if, for every linear order of the vertex set, there is some edge oriented consistently with the linear order. We find bounds on the minimum number of edges in a hypergraph with Property O.

scholarly journals G-Intersection Theorems for Matchings and Other Graphs

2008 ◽  
Vol 17 (4) ◽  
pp. 559-575
Author(s):  
J. ROBERT JOHNSON ◽  
JOHN TALBOT
Keyword(s):  
Phase Transition ◽  
Positive Root ◽  
Sufficient Condition ◽  
Matching Number ◽  
Sharp Transition ◽  
Intersecting Family ◽  
Vertex Set ◽  
Almost All ◽  
Sharp Phase Transition

If G is a graph with vertex set [n] then $\mathcal{A}\subseteq 2^{[n]}$ is G-intersecting if, for all $A,B\in \mathcal{A}$, either A ∩ B ≠ ∅ or there exist a ∈ A and b ∈ B such that a ~Gb.The question of how large a k-uniform G-intersecting family can be was first considered by Bohman, Frieze, Ruszinkó and Thoma [2], who identified two natural candidates for the extrema depending on the relative sizes of k and n and asked whether there is a sharp phase transition between the two. Our first result shows that there is a sharp transition and characterizes the extremal families when G is a matching. We also give an example demonstrating that other extremal families can occur.Our second result gives a sufficient condition for the largest G-intersecting family to contain almost all k-sets. In particular we show that if Cn is the n-cycle and k > αn + o(n), where α = 0.266. . . is the smallest positive root of (1 − x)3(1 + x) = 1/2, then the largest Cn-intersecting family has size $(1-o(1))\binom{n}{k}$.Finally we consider the non-uniform problem, and show that in this case the size of the largest G-intersecting family depends on the matching number of G.

hClique: An exact algorithm for maximum clique problem in uniform hypergraphs

2017 ◽  
Vol 09 (06) ◽  
pp. 1750078 ◽  
Author(s):  
Jose Torres-Jimenez ◽  
Jose Carlos Perez-Torres ◽  
Gildardo Maldonado-Martinez
Keyword(s):  
Maximum Clique ◽  
Exact Algorithm ◽  
Maximum Clique Problem ◽  
Uniform Hypergraph ◽  
Uniform Hypergraphs ◽  
Vertex Set ◽  
Clique Problem

A hypergraph [Formula: see text] with vertex set [Formula: see text] and edge set [Formula: see text] differs from a graph in that an edge can connect more than two vertices. An r-uniform hypergraph [Formula: see text] is a hypergraph with hyperedges of size [Formula: see text]. For an r-uniform hypergraph [Formula: see text], an r-uniform clique is a subset [Formula: see text] of [Formula: see text] such as every subset of [Formula: see text] elements of [Formula: see text] belongs to [Formula: see text]. We present hClique, an exact algorithm to find a maximum r-uniform clique for [Formula: see text]-uniform graphs. In order to evidence the performance of hClique, 32 random [Formula: see text]-graphs were solved.

scholarly journals Turán Problems on Non-Uniform Hypergraphs

10.37236/3901 ◽  
2014 ◽  
Vol 21 (4) ◽  
Author(s):  
J. Travis Johnston ◽  
Linyuan Lu
Keyword(s):  
Recent Work ◽  
Blow Up ◽  
Uniform Hypergraph ◽  
Expected Number ◽  
Uniform Hypergraphs ◽  
Turan Problems ◽  
Vertex Set ◽  
Lubell Function

A non-uniform hypergraph $H=(V,E)$ consists of a vertex set $V$ and an edge set $E\subseteq 2^V$; the edges in $E$ are not required to all have the same cardinality. The set of all cardinalities of edges in $H$ is denoted by $R(H)$, the set of edge types. For a fixed hypergraph $H$, the Turán density $\pi(H)$ is defined to be $\lim_{n\to\infty}\max_{G_n}h_n(G_n)$, where the maximum is taken over all $H$-free hypergraphs $G_n$ on $n$ vertices satisfying $R(G_n)\subseteq R(H)$, and $h_n(G_n)$, the so called Lubell function, is the expected number of edges in $G_n$ hit by a random full chain. This concept, which generalizes  the Turán density of $k$-uniform hypergraphs, is motivated by recent work on extremal poset problems.  The details connecting these two areas will be revealed in the end of this paper.Several properties of Turán density, such as supersaturation, blow-up, and suspension, are generalized from uniform hypergraphs to non-uniform hypergraphs. Other questions such as "Which hypergraphs are degenerate?" are more complicated and don't appear to generalize well. In addition, we completely determine the Turán densities of $\{1,2\}$-hypergraphs.

scholarly journals Cyclic partitions of complete uniform hypergraphs

10.37236/390 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Artur Szymański ◽  
A. Paweł Wojda
Keyword(s):  
Sufficient Condition ◽  
Uniform Hypergraph ◽  
Necessary And Sufficient Condition ◽  
Uniform Hypergraphs ◽  
Necessary And Sufficient ◽  
Sigma E

By $K^{(k)}_n$ we denote the complete $k$-uniform hypergraph of order $n$, $1\le k \le n-1$, i.e. the hypergraph with the set $V_n=\{ 1,2,...,n\}$ of vertices and the set $V_n \choose k$ of edges. If there exists a permutation $\sigma$ of the set $V_n$ such that $\{ E,\sigma (E),..., \sigma^{q-1}(E) \}$ is a partition of the set $V_n \choose k$ then we call it cyclic $q$-partition of $K^{(k)}_n$ and $\sigma$ is said to be a $(q,k)$-complementing. In the paper, for arbitrary integers $k,q$ and $n$, we give a necessary and sufficient condition for a permutation to be $(q,k)$-complementing permutation of $K^{(k)}_n$. By $\tilde{K}_n$ we denote the hypergraph with the set of vertices $V_n$ and the set of edges $2^{V_n} - \{ \emptyset , V_n \}$. If there is a permutation $\sigma$ of $V_n$ and a set $E \subset 2^{V_n} - \{ \emptyset , V_n \}$ such that $\{ E, \sigma (E),..., \sigma^{p-1}(E) \}$ is a $p$-partition of $ 2^{V_n} - \{ \emptyset , V_n \}$ then we call it a cyclic $p$-partition of $K_n$ and we say that $\sigma$ is $p$-complementing. We prove that $\tilde{K}_n$ has a cyclic $p$-partition if and only if $p$ is prime and $n$ is a power of $p$ (and $n > p$). Moreover, any $p$-complementing permutation is cyclic.

scholarly journals Cyclic partitions of complete nonuniform hypergraphs and complete multipartite hypergraphs

2013 ◽  
Vol Vol. 15 no. 2 (Combinatorics) ◽  
Author(s):  
Shonda Gosselin ◽  
Andrzej Szymański ◽  
Adam Pawel Wojda
Keyword(s):  
Positive Integer ◽  
Nonempty Subset ◽  
Sufficient Conditions ◽  
Sufficient Condition ◽  
Uniform Hypergraph ◽  
Necessary And Sufficient Condition ◽  
Vertex Set ◽  
International Audience ◽  
Complete Multipartite ◽  
Necessary And Sufficient

Combinatorics International audience A \em cyclic q-partition of a hypergraph (V,E) is a partition of the edge set E of the form \F,F^θ,F^θ², \ldots, F^θ^q-1\ for some permutation θ of the vertex set V. Let Vₙ = \ 1,2,\ldots,n\. For a positive integer k, Vₙ\choose k denotes the set of all k-subsets of Vₙ. For a nonempty subset K of V_n-1, we let \mathcalKₙ^(K) denote the hypergraph ≤ft(Vₙ, \bigcup_k∈ K Vₙ\choose k\right). In this paper, we find a necessary and sufficient condition on n, q and k for the existence of a cyclic q-partition of \mathcalKₙ^(V_k). In particular, we prove that if p is prime then there is a cyclic p^α-partition of \mathcalK^(Vₖ)ₙ if and only if p^α + β divides n, where β = \lfloor \logₚ k\rfloor. As an application of this result, we obtain two sufficient conditions on n₁,n₂,\ldots,n_t, k, α and a prime p for the existence of a cyclic p^α-partition of the complete t-partite k-uniform hypergraph \mathcal K^(k)_n₁,n₂,\ldots,n_t.

scholarly journals Rainbow Hamilton Cycles in Uniform Hypergraphs

10.37236/2055 ◽  
2012 ◽  
Vol 19 (1) ◽  
Author(s):  
Andrzej Dudek ◽  
Alan Frieze ◽  
Andrzej Ruciński
Keyword(s):  
Maximum Degree ◽  
Hamilton Cycle ◽  
Uniform Hypergraph ◽  
Hamilton Cycles ◽  
Lovász Local Lemma ◽  
Uniform Hypergraphs ◽  
Local Lemma ◽  
Large N ◽  
Different Color ◽  
Open Question

Let $K_n^{(k)}$ be the complete $k$-uniform hypergraph, $k\ge3$, and let $\ell$ be an integer such that $1\le \ell\le k-1$ and $k-\ell$ divides $n$. An $\ell$-overlapping Hamilton cycle in $K_n^{(k)}$ is a spanning subhypergraph $C$ of  $K_n^{(k)}$  with $n/(k-\ell)$ edges and such that for some cyclic ordering of the vertices each edge of $C$ consists of $k$ consecutive vertices and every pair of adjacent edges in $C$ intersects in precisely $\ell$ vertices.We show that, for some constant $c=c(k,\ell)$ and sufficiently large $n$, for every coloring (partition) of the edges of $K_n^{(k)}$ which uses arbitrarily many colors but no color appears more than $cn^{k-\ell}$ times, there exists a rainbow $\ell$-overlapping Hamilton cycle $C$, that is every edge of $C$ receives a different color. We also prove that, for some constant $c'=c'(k,\ell)$ and sufficiently large $n$, for every coloring of the edges of $K_n^{(k)}$ in which the maximum degree of the subhypergraph induced by any single color is bounded by $c'n^{k-\ell}$,  there exists a properly colored $\ell$-overlapping Hamilton cycle $C$, that is every two adjacent edges receive different colors. For $\ell=1$, both results are (trivially) best possible up to the constants. It is an open question if our results are also optimal for $2\le\ell\le k-1$.The proofs  rely on a version of the Lovász Local Lemma and incorporate some ideas from Albert, Frieze, and Reed.

scholarly journals Nearly-Regular Hypergraphs and Saturation of Berge Stars

10.37236/8363 ◽  
2019 ◽  
Vol 26 (4) ◽  
Author(s):  
Bethany Austhof ◽  
Sean English
Keyword(s):  
Uniform Hypergraph ◽  
Uniform Hypergraphs ◽  
Vertex Set ◽  
Saturation Number

Given a graph $G$, we say a $k$-uniform hypergraph $H$ on the same vertex set contains a Berge-$G$ if there exists an injection $\phi:E(G)\to E(H)$ such that $e\subseteq\phi(e)$ for each edge $e\in E(G)$. A hypergraph $H$ is Berge-$G$-saturated if $H$ does not contain a Berge-$G$, but adding any edge to $H$ creates a Berge-$G$. The saturation number for Berge-$G$, denoted $\mathrm{sat}_k(n,\text{Berge-}G)$ is the least number of edges in a $k$-uniform hypergraph that is Berge-$G$-saturated. We determine exactly the value of the saturation numbers for Berge stars. As a tool for our main result, we also prove the existence of nearly-regular $k$-uniform hypergraphs, or $k$-uniform hypergraphs in which every vertex has degree $r$ or $r-1$ for some $r\in \mathbb{Z}$, and less than $k$ vertices have degree $r-1$. 

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