scholarly journals Hypercycle Systems of 5-Cycles in Complete 3-Uniform Hypergraphs

Mathematics ◽  
2021 ◽  
Vol 9 (5) ◽  
pp. 484
Author(s):  
Anita Keszler ◽  
Zsolt Tuza
Keyword(s):  
Element Subset ◽  
Uniform Hypergraphs ◽  
Vertex Set ◽  
Recursive Constructions

In this paper, we consider the problem of constructing hypercycle systems of 5-cycles in complete 3-uniform hypergraphs. A hypercycle system C(r,k,v) of order v is a collection of r-uniform k-cycles on a v-element vertex set, such that each r-element subset is an edge in precisely one of those k-cycles. We present cyclic hypercycle systems C(3,5,v) of orders v=25,26,31,35,37,41,46,47,55,56, a highly symmetric construction for v=40, and cyclic 2-split constructions of orders 32,40,50,52. As a consequence, all orders v≤60 permitted by the divisibility conditions admit a C(3,5,v) system. New recursive constructions are also introduced.

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 Harmonious coloring of uniform hypergraphs

2016 ◽  
Vol 10 (1) ◽  
pp. 73-87 ◽  
Author(s):  
Bartłomiej Bosek ◽  
Sebastian Czerwiński ◽  
Jarosław Grytczuk ◽  
Paweł Rzążewski
Keyword(s):  
Maximum Degree ◽  
Vertex Coloring ◽  
Uniform Hypergraph ◽  
Element Subset ◽  
Multiplicative Constant ◽  
Uniform Hypergraphs ◽  
Local Lemma ◽  
Novel Method ◽  
Entropy Compression

A harmonious coloring of a k-uniform hypergraph H is a vertex coloring such that no two vertices in the same edge share the same color, and each k-element subset of colors appears on at most one edge. The harmonious number h(H) is the least number of colors needed for such a coloring. We prove that k-uniform hypergraphs of bounded maximum degree ? satisfy h(H) = O(k?k!m), where m is the number of edges in H which is best possible up to a multiplicative constant. Moreover, for every fixed ?, this constant tends to 1 with k ? ?. We use a novel method, called entropy compression, that emerged from the algorithmic version of the Lov?sz Local Lemma due to Moser and Tardos.

scholarly journals Extremal problems for convex geometric hypergraphs and ordered hypergraphs

2020 ◽  
pp. 1-19
Author(s):  
Zoltán Füredi ◽  
Tao Jiang ◽  
Alexandr Kostochka ◽  
Dhruv Mubayi ◽  
Jacques Verstraëte
Keyword(s):  
Extremal Function ◽  
Combinatorial Geometry ◽  
Extremal Problems ◽  
Geometric Graphs ◽  
Uniform Hypergraphs ◽  
Vertex Set ◽  
Order Of Magnitude ◽  
The Extremal Function ◽  
Geometric Hypergraphs ◽  
Rich History

Abstract An ordered hypergraph is a hypergraph whose vertex set is linearly ordered, and a convex geometric hypergraph is a hypergraph whose vertex set is cyclically ordered. Extremal problems for ordered and convex geometric graphs have a rich history with applications to a variety of problems in combinatorial geometry. In this paper, we consider analogous extremal problems for uniform hypergraphs, and determine the order of magnitude of the extremal function for various ordered and convex geometric paths and matchings. Our results generalize earlier works of Braı–Károlyi–Valtr, Capoyleas–Pach, and Aronov–Dujmovič–Morin–Ooms-da Silveira. We also provide a new variation of the Erdős-Ko-Rado theorem in the ordered setting.

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

2019 ◽  
Vol 28 (06) ◽  
pp. 840-870 ◽  
Author(s):  
Wei Gao ◽  
Jie Han ◽  
Yi Zhao
Keyword(s):  
Error Term ◽  
Uniform Hypergraphs ◽  
Turán Number ◽  
Vertex Set ◽  
Vertex Disjoint

AbstractGiven two k-graphs (k-uniform hypergraphs) F and H, a perfect F-tiling (or F-factor) in H is a set of vertex-disjoint copies of F that together cover the vertex set of H. For all complete k-partite k-graphs K, Mycroft proved a minimum codegree condition that guarantees a K-factor in an n-vertex k-graph, which is tight up to an error term o(n). In this paper we improve the error term in Mycroft’s result to a sublinear term that relates to the Turán number of K when the differences of the sizes of the vertex classes of K are 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 tiling K(k)(1, … , 1, 2) and tiling loose cycles, thus generalizing the results of Czygrinow, DeBiasio and Nagle, and of Czygrinow, respectively.

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 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 Spanning trees in random regular uniform hypergraphs

2021 ◽  
pp. 1-25
Author(s):  
Catherine Greenhill ◽  
Mikhail Isaev ◽  
Gary Liang
Keyword(s):  
Asymptotic Distribution ◽  
Spanning Tree ◽  
Spanning Trees ◽  
Threshold Value ◽  
Regular Graphs ◽  
Cubic Graphs ◽  
Trivial Case ◽  
Uniform Hypergraphs ◽  
Vertex Set ◽  
Number Of Spanning Trees

Abstract Let $${{\mathcal G}_{n,r,s}}$$ denote a uniformly random r-regular s-uniform hypergraph on the vertex set {1, 2, … , n}. We establish a threshold result for the existence of a spanning tree in $${{\mathcal G}_{n,r,s}}$$ , restricting to n satisfying the necessary divisibility conditions. Specifically, we show that when s ≥ 5, there is a positive constant ρ(s) such that for any r ≥ 2, the probability that $${{\mathcal G}_{n,r,s}}$$ contains a spanning tree tends to 1 if r > ρ(s), and otherwise this probability tends to zero. The threshold value ρ(s) grows exponentially with s. As $${{\mathcal G}_{n,r,s}}$$ is connected with probability that tends to 1, this implies that when r ≤ ρ(s), most r-regular s-uniform hypergraphs are connected but have no spanning tree. When s = 3, 4 we prove that $${{\mathcal G}_{n,r,s}}$$ contains a spanning tree with probability that tends to 1, for any r ≥ 2. Our proof also provides the asymptotic distribution of the number of spanning trees in $${{\mathcal G}_{n,r,s}}$$ for all fixed integers r, s ≥ 2. Previously, this asymptotic distribution was only known in the trivial case of 2-regular graphs, or for cubic graphs.

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