scholarly journals A Generalization of Erdős' Matching Conjecture

10.37236/7420 ◽  
2018 ◽  
Vol 25 (2) ◽  
Author(s):  
Christos Pelekis ◽  
Israel Rocha
Keyword(s):  
Positive Integer ◽  
General Problem ◽  
Matching Number ◽  
Uniform Hypergraph ◽  
Maximum Cardinality ◽  
Intersection Problem ◽  
Extremal Hypergraph ◽  
Candidate Set

Let $\mathcal{H}=(V,\mathcal{E})$ be an $r$-uniform hypergraph on $n$ vertices and fix a positive integer $k$ such that $1\le k\le r$. A $k$-matching of $\mathcal{H}$ is a collection of edges $\mathcal{M}\subset \mathcal{E}$ such that every subset of $V$ whose cardinality equals $k$ is contained in at most one element of $\mathcal{M}$. The $k$-matching number of $\mathcal{H}$ is the maximum cardinality of a $k$-matching. A well-known problem, posed by Erdős, asks for the maximum number of edges in an $r$-uniform hypergraph under constraints on its $1$-matching number. In this article we investigate the more general problem of determining the maximum number of edges in an $r$-uniform hypergraph on $n$ vertices subject to the constraint that its $k$-matching number is strictly less than $a$. The problem can also be seen as a generalization of the well-known $k$-intersection problem. We propose candidate hypergraphs for the solution of this problem, and show that the extremal hypergraph is among this candidate set when $n\ge 4r\binom{r}{k}^2\cdot a$.

Decomposing Complete 3-Uniform Hypergraph into 5-Cycles

2014 ◽  
Vol 672-674 ◽  
pp. 1935-1939
Author(s):  
Guan Ru Li ◽  
Yi Ming Lei ◽  
Jirimutu
Keyword(s):  
Positive Integer ◽  
Hamiltonian Cycles ◽  
Uniform Hypergraph ◽  
Design Algorithm ◽  
Uniform Hypergraphs ◽  
Edge Partition ◽  
Definition Of

About the Katona-Kierstead definition of a Hamiltonian cycles in a uniform hypergraph, a decomposition of complete k-uniform hypergraph Kn(k) into Hamiltonian cycles studied by Bailey-Stevens and Meszka-Rosa. For n≡2,4,5 (mod 6), we design algorithm for decomposing the complete 3-uniform hypergraphs into Hamiltonian cycles by using the method of edge-partition. A decomposition of Kn(3) into 5-cycles has been presented for all admissible n≤17, and for all n=4m +1, m is a positive integer. In general, the existence of a decomposition into 5-cycles remains open. In this paper, we use the method of edge-partition and cycle sequence proposed by Jirimutu and Wang. We find a decomposition of K20(3) into 5-cycles.

scholarly journals The Size of a Hypergraph and its Matching Number

2012 ◽  
Vol 21 (3) ◽  
pp. 442-450 ◽  
Author(s):  
HAO HUANG ◽  
PO-SHEN LOH ◽  
BENNY SUDAKOV
Keyword(s):  
Matching Number ◽  
Uniform Hypergraph ◽  
Edge Matching ◽  
Turan Problem ◽  
Turán Problem

More than forty years ago, Erdős conjectured that for any $t \leq \frac{n}{k}$, every k-uniform hypergraph on n vertices without t disjoint edges has at most max${\binom{kt-1}{k}, \binom{n}{k}-\binom{n-t+1}{k}\}$ edges. Although this appears to be a basic instance of the hypergraph Turán problem (with a t-edge matching as the excluded hypergraph), progress on this question has remained elusive. In this paper, we verify this conjecture for all $t < \frac{n}{3k^2}$. This improves upon the best previously known range $t = O\bigl(\frac{n}{k^3}\bigr)$, which dates back to the 1970s.

scholarly journals Decomposing complete 3-uniform hypergraph K_{n}^{(3)} into 7-cycles

2019 ◽  
Vol 39 (3) ◽  
pp. 383-393
Author(s):  
Meihua Meihua ◽  
Meiling Guan ◽  
Jirimutu Jirimutu
Keyword(s):  
Positive Integer ◽  
Hamiltonian Cycle ◽  
Hamiltonian Cycles ◽  
Uniform Hypergraph ◽  
Uniform Hypergraphs ◽  
Edge Partition ◽  
Definition Of

We use the Katona-Kierstead definition of a Hamiltonian cycle in a uniform hypergraph. A decomposition of complete \(k\)-uniform hypergraph \(K^{(k)}_{n}\) into Hamiltonian cycles was studied by Bailey-Stevens and Meszka-Rosa. For \(n\equiv 2,4,5\pmod 6\), we design an algorithm for decomposing the complete 3-uniform hypergraphs into Hamiltonian cycles by using the method of edge-partition. A decomposition of \(K^{(3)}_{n}\) into 5-cycles has been presented for all admissible \(n\leq17\), and for all \(n=4^{m}+1\) when \(m\) is a positive integer. In general, the existence of a decomposition into 5-cycles remains open. In this paper, we show if \(42~|~(n-1)(n-2)\) and if there exist \(\lambda=\frac{(n-1)(n-2)}{42}\) sequences \((k_{i_{0}},k_{i_{1}},\ldots,k_{i_{6}})\) on \(D_{all}(n)\), then \(K^{(3)}_{n}\) can be decomposed into 7-cycles. We use the method of edge-partition and cycle sequence. We find a decomposition of \(K^{(3)}_{37}\) and \(K^{(3)}_{43}\) into 7-cycles.

scholarly journals Self-complementing permutations of k-uniform hypergraphs

2009 ◽  
Vol Vol. 11 no. 1 (Graph and Algorithms) ◽  
Author(s):  
Artur Szymański ◽  
Adam Pawel Wojda
Keyword(s):  
Positive Integer ◽  
Uniform Hypergraph ◽  
Uniform Hypergraphs ◽  
Unique Integer ◽  
International Audience ◽  
Sigma E

Graphs and Algorithms International audience A k-uniform hypergraph H = ( V; E) is said to be self-complementary whenever it is isomorphic with its complement (H) over bar = ( V; ((V)(k)) - E). Every permutation sigma of the set V such that sigma(e) is an edge of (H) over bar if and only if e is an element of E is called self-complementing. 2-self-comlementary hypergraphs are exactly self complementary graphs introduced independently by Ringel ( 1963) and Sachs ( 1962). <br> For any positive integer n we denote by lambda(n) the unique integer such that n = 2(lambda(n)) c, where c is odd. <br> In the paper we prove that a permutation sigma of [1, n] with orbits O-1,..., O-m O m is a self-complementing permutation of a k-uniform hypergraph of order n if and only if there is an integer l >= 0 such that k = a2(l) + s, a is odd, 0 <= s <= 2(l) and the following two conditions hold: <br> (i)n = b2(l+1) + r,r is an element of {0,..., 2(l) - 1 + s}, and <br> (ii) Sigma(i:lambda(vertical bar Oi vertical bar)<= l) vertical bar O-i vertical bar <= r. <br> For k = 2 this result is the very well known characterization of self-complementing permutation of graphs given by Ringel and Sachs.

Linear Turán Numbers of Linear Cycles and Cycle-Complete Ramsey Numbers

2017 ◽  
Vol 27 (3) ◽  
pp. 358-386 ◽  
Author(s):  
CLAYTON COLLIER-CARTAINO ◽  
NATHAN GRABER ◽  
TAO JIANG
Keyword(s):  
Positive Integer ◽  
Positive Constant ◽  
Uniform Hypergraph ◽  
Ramsey Numbers ◽  
Turán Number ◽  
Formula Group ◽  
Image Position ◽  
Linear Hypergraphs

Anr-uniform hypergraph is called anr-graph. A hypergraph islinearif every two edges intersect in at most one vertex. Given a linearr-graphHand a positive integern, thelinear Turán numberexL(n,H) is the maximum number of edges in a linearr-graphGthat does not containHas a subgraph. For each ℓ ≥ 3, letCrℓdenote ther-uniform linear cycle of length ℓ, which is anr-graph with edgese1, . . .,eℓsuch that, for alli∈ [ℓ−1], |ei∩ei+1|=1, |eℓ∩e1|=1 andei∩ej= ∅ for all other pairs {i,j},i≠j. For allr≥ 3 and ℓ ≥ 3, we show that there exists a positive constantc=cr,ℓ, depending onlyrand ℓ, such that exL(n,Crℓ) ≤cn1+1/⌊ℓ/2⌋. This answers a question of Kostochka, Mubayi and Verstraëte [30]. For even ℓ, our result extends the result of Bondy and Simonovits [7] on the Turán numbers of even cycles to linear hypergraphs.Using our results on linear Turán numbers, we also obtain bounds on the cycle-complete hypergraph Ramsey numbers. We show that there are positive constantsa=am,randb=bm,r, depending only onmandr, such that\begin{equation} R(C^r_{2m}, K^r_t)\leq a \Bigl(\frac{t}{\ln t}\Bigr)^{{m}/{(m-1)}} \quad\text{and}\quad R(C^r_{2m+1}, K^r_t)\leq b t^{{m}/{(m-1)}}. \end{equation}

Cover Product and Betti Polynomial of Graphs

2015 ◽  
Vol 58 (2) ◽  
pp. 320-333
Author(s):  
Aurora Llamas ◽  
Josá Martínez–Bernal
Keyword(s):  
Positive Integer ◽  
Bipartite Graph ◽  
Betti Numbers ◽  
Recursive Formula ◽  
Bipartite Graphs ◽  
Matching Number ◽  
Edge Ideal ◽  
Graded Betti Numbers ◽  
Vertex Set ◽  
Chordal Bipartite Graphs

AbstractThe cover product of disjoint graphs G and H with fixed vertex covers C(G) and C(H), is the graphwith vertex set V(G) ∪ V(H) and edge setWe describe the graded Betti numbers of GeH in terms of those of. As applications we obtain: (i) For any positive integer k there exists a connected bipartite graph G such that reg R/I(G) = μS(G) + k, where, I(G) denotes the edge ideal of G, reg R/I(G) is the Castelnuovo–Mumford regularity of R/I(G) and μS(G) is the induced or strong matching number of G; (ii)The graded Betti numbers of the complement of a tree depends only upon its number of vertices; (iii)The h-vector of R/I(G e H) is described in terms of the h-vectors of R/I(G) and R/I(H). Furthermore, in a diòerent direction, we give a recursive formula for the graded Betti numbers of chordal bipartite graphs.

scholarly journals On the Minimum Number of Monochromatic Generalized Schur Triples

10.37236/6490 ◽  
2017 ◽  
Vol 24 (2) ◽  
Author(s):  
Thotsaporn Thanatipanonda ◽  
Elaine Wong
Keyword(s):  
Positive Integer ◽  
General Problem ◽  
Original Proof ◽  
Minimum Number ◽  
Fixed Positive Integer

The solution to the problem of finding the minimum number of monochromatic triples $(x,y,x+ay)$ with $a\geq 2$ being a fixed positive integer over any 2-coloring of $[1,n]$ was conjectured by Butler, Costello, and Graham (2010) and Thanathipanonda (2009). We solve this problem using a method based on Datskovsky's proof (2003) on the minimum number of monochromatic Schur triples $(x,y,x+y)$. We do this by exploiting the combinatorial nature of the original proof and adapting it to the general problem.

scholarly journals Squarefree Powers of Edge Ideals of Forests

2021 ◽  
Vol 28 (2) ◽  
Author(s):  
Nursel Erey ◽  
Takayuki Hibi
Keyword(s):  
Positive Integer ◽  
Upper Bound ◽  
Matching Number ◽  
Edge Ideal ◽  
Combinatorial Formula ◽  
Edge Ideals ◽  
Linear Resolution

Let $I(G)^{[k]}$ denote the $k$th squarefree power of the edge ideal of $G$. When $G$ is a forest, we provide a sharp upper bound for the regularity of $I(G)^{[k]}$ in terms of the $k$-admissable matching number of $G$. For any positive integer $k$, we classify all forests $G$ such that $I(G)^{[k]}$ has linear resolution. We also give a combinatorial formula for the regularity of $I(G)^{[2]}$ for any forest $G$.

scholarly journals On Upper Transversals in 3-Uniform Hypergraphs

10.37236/7267 ◽  
2018 ◽  
Vol 25 (4) ◽  
Author(s):  
Michael A. Henning ◽  
Anders Yeo
Keyword(s):  
Nonempty Intersection ◽  
Uniform Hypergraph ◽  
Maximum Cardinality ◽  
Uniform Hypergraphs ◽  
Transversal Number

A set $S$ of vertices in a hypergraph $H$ is a transversal if it has a nonempty intersection with every edge of $H$. The upper transversal number $\Upsilon(H)$ of $H$ is the maximum cardinality of a minimal transversal in $H$. We show that if $H$ is a connected $3$-uniform hypergraph of order $n$, then $\Upsilon(H) > 1.4855 \sqrt[3]{n} - 2$. For $n$ sufficiently large, we construct infinitely many connected $3$-uniform hypergraphs, $H$, of order~$n$ satisfying $\Upsilon(H) < 2.5199 \sqrt[3]{n}$. We conjecture that $\displaystyle{\sup_{n \to \infty}  \, \left( \inf  \frac{ \Upsilon(H) }{ \sqrt[3]{n} } \right) = \sqrt[3]{16} }$, where the infimum is taken over all connected $3$-uniform hypergraphs $H$ of order $n$.

scholarly journals Vertex-Transitive $q$-Complementary Uniform Hypergraphs

10.37236/587 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Shonda Gosselin
Keyword(s):  
Positive Integer ◽  
Prime Number ◽  
Prime Order ◽  
Necessary Conditions ◽  
Transitive Permutation Group ◽  
Uniform Hypergraph ◽  
Complete Group ◽  
Large Sets ◽  
Uniform Hypergraphs ◽  
Vertex Transitive

For a positive integer $q$, a $k$-uniform hypergraph $X=(V,E)$ is $q$-complementary if there exists a permutation $\theta$ on $V$ such that the sets $E, E^{\theta}, E^{\theta^2},\ldots, E^{\theta^{q-1}}$ partition the set of $k$-subsets of $V$. The permutation $\theta$ is called a $q$-antimorphism of $X$. The well studied self-complementary uniform hypergraphs are 2-complementary. For an integer $n$ and a prime $p$, let $n_{(p)}=\max\{i:p^i \text{divides} n\}$. In this paper, we prove that a vertex-transitive $q$-complementary $k$-hypergraph of order $n$ exists if and only if $n^{n_{(p)}}\equiv 1 (\bmod q^{\ell+1})$ for every prime number $p$, in the case where $q$ is prime, $k = bq^\ell$ or $k=bq^{\ell}+1$ for a positive integer $b < k$, and $n\equiv 1(\bmod q^{\ell+1})$. We also find necessary conditions on the order of these structures when they are $t$-fold-transitive and $n\equiv t (\bmod q^{\ell+1})$, for $1\leq t < k$, in which case they correspond to large sets of isomorphic $t$-designs. Finally, we use group theoretic results due to Burnside and Zassenhaus to determine the complete group of automorphisms and $q$-antimorphisms of these hypergraphs in the case where they have prime order, and then use this information to write an algorithm to generate all of these objects. This work extends previous, analagous results for vertex-transitive self-complementary uniform hypergraphs due to Muzychuk, Potočnik, Šajna, and the author. These results also extend the previous work of Li and Praeger on decomposing the orbitals of a transitive permutation group.

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