scholarly journals Maximum packings of the λ-fold complete 3-uniform hypergraph with loose 3-cycles

2020 ◽  
Vol 40 (2) ◽  
pp. 209-225
Author(s):  
Ryan C. Bunge ◽  
Dontez Collins ◽  
Daryl Conko-Camel ◽  
Saad I. El-Zanati ◽  
Rachel Liebrecht ◽  
...  
Keyword(s):  
Uniform Hypergraph ◽  
Positive Integers ◽  
Maximum Packings

It is known that the 3-uniform loose 3-cycle decomposes the complete 3-uniform hypergraph of order \(v\) if and only if \(v \equiv 0, 1,\text{ or }2 (\operatorname{mod} 9)\). For all positive integers \(\lambda\) and \(v\), we find a maximum packing with loose 3-cycles of the \(\lambda\)-fold complete 3-uniform hypergraph of order \(v\). We show that, if \(v \geq 6\), such a packing has a leave of two or fewer edges.

scholarly journals Vertex Degree Sums for Matchings in 3-Uniform Hypergraphs

10.37236/8627 ◽  
2019 ◽  
Vol 26 (4) ◽  
Author(s):  
Yi Zhang ◽  
Yi Zhao ◽  
Mei Lu
Keyword(s):  
Vertex Degree ◽  
Uniform Hypergraph ◽  
Degree Sum ◽  
Uniform Hypergraphs ◽  
Positive Integers ◽  
Degree Sum Condition

Let $n, s$ be positive integers such that $n$ is sufficiently large and $s\le n/3$. Suppose $H$ is a 3-uniform hypergraph of order $n$ without isolated vertices. If $\deg(u)+\deg(v) > 2(s-1)(n-1)$ for any two vertices $u$ and $v$ that are contained in some edge of $H$, then $H$ contains a matching of size $s$. This degree sum condition is best possible and confirms a conjecture of the authors [Electron. J. Combin. 25 (3), 2018], who proved the case when $s= n/3$.

scholarly journals An extremal problem in hypergraph theory (II)

1981 ◽  
Vol 31 (2) ◽  
pp. 129-135 ◽  
Author(s):  
H. L. Abbott ◽  
D. Hanson ◽  
A. C. Liu
Keyword(s):  
Extremal Problem ◽  
Independent Set ◽  
Uniform Hypergraph ◽  
Hypergraph Theory ◽  
Positive Integers

AbstractLet t, m > 2 and p > 2 be positive integers and denote by N(t, m, p) the largest integer for which there exists a t-uniform hypergraph with N (not necessarily distinct) edges and having no independent set of edges of size m and no vertex of degree exceeding p. In this paper we complete the determination of N(t, m, 3) and obtain some new bounds on N(t, 2, p).

scholarly journals A linear upper bound in zero-sum Ramsey theory

1994 ◽  
Vol 17 (3) ◽  
pp. 609-612 ◽  
Author(s):  
Yair Caro
Keyword(s):  
Upper Bound ◽  
Ramsey Theory ◽  
Ramsey Number ◽  
Uniform Hypergraph ◽  
Zero Sum ◽  
Positive Integers

Letn,randkbe positive integers such thatk|(nr). There exists a constantc(k,r)such that for fixedkandrand for every groupAof orderkR(Knr,A)≤n+c(k,r),whereR(Knr,A)is the zero-sum Ramsey number introduced by Bialostocki and Dierker [1], andKnris the completer-uniform hypergraph onn-vertices.

scholarly journals Rainbow $H$-Factors of Complete $s$-Uniform $r$-Partite Hypergraphs

10.37236/901 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Ailian Chen ◽  
Fuji Zhang ◽  
Hao Li
Keyword(s):  
Chromatic Number ◽  
Edge Coloring ◽  
Uniform Hypergraph ◽  
Vertex Partition ◽  
Proper Edge Coloring ◽  
Vertex Set ◽  
Positive Integers

We say a $s$-uniform $r$-partite hypergraph is complete, if it has a vertex partition $\{V_1,V_2,...,V_r\}$ of $r$ classes and its hyperedge set consists of all the $s$-subsets of its vertex set which have at most one vertex in each vertex class. We denote the complete $s$-uniform $r$-partite hypergraph with $k$ vertices in each vertex class by ${\cal T}_{s,r}(k)$. In this paper we prove that if $h,\ r$ and $s$ are positive integers with $2\leq s\leq r\leq h$ then there exists a constant $k=k(h,r,s)$ so that if $H$ is an $s$-uniform hypergraph with $h$ vertices and chromatic number $\chi(H)=r$ then any proper edge coloring of ${\cal T}_{s,r}(k)$ has a rainbow $H$-factor.

Language, procedures, and the non-perceptual origin of natural number concepts

2016 ◽  
Author(s):  
David Barner
Keyword(s):  
General Framework ◽  
Ad Hoc ◽  
Building Blocks ◽  
Number Words ◽  
Linguistic Markers ◽  
Logical Foundations ◽  
Large Numbers ◽  
Perceptual Representations ◽  
Positive Integers ◽  
Perceptual Systems

Perceptual representations – e.g., of objects or approximate magnitudes –are often invoked as building blocks that children combine with linguisticsymbols when they acquire the positive integers. Systems of numericalperception are either assumed to contain the logical foundations ofarithmetic innately, or to supply the basis for their induction. Here Ipropose an alternative to this general framework, and argue that theintegers are not learned from perceptual systems, but instead arise toexplain perception as part of language acquisition. Drawing oncross-linguistic data and developmental data, I show that small numbers(1-4) and large numbers (~5+) arise both historically and in individualchildren via entirely distinct mechanisms, constituting independentlearning problems, neither of which begins with perceptual building blocks.Specifically, I propose that children begin by learning small numbers(i.e., *one, two, three*) using the same logical resources that supportother linguistic markers of number (e.g., singular, plural). Several yearslater, children discover the logic of counting by inferring the logicalrelations between larger number words from their roles in blind countingprocedures, and only incidentally associate number words with perception ofapproximate magnitudes, in an *ad hoc* and highly malleable fashion.Counting provides a form of explanation for perception but is not causallyderived from perceptual systems.

Short Generating Functions for some Semigroup Algebras

10.37236/1729 ◽  
2003 ◽  
Vol 10 (1) ◽  
Author(s):  
Graham Denham
Keyword(s):  
Rational Function ◽  
Generating Functions ◽  
Hilbert Series ◽  
Simple Expression ◽  
Arbitrary Field ◽  
Graded Algebra ◽  
Semigroup Algebras ◽  
Positive Integers

Let $a_1,\ldots,a_n$ be distinct, positive integers with $(a_1,\ldots,a_n)=1$, and let k be an arbitrary field. Let $H(a_1,\ldots,a_n;z)$ denote the Hilbert series of the graded algebra k$[t^{a_1},t^{a_2},\ldots,t^{a_n}]$. We show that, when $n=3$, this rational function has a simple expression in terms of $a_1,a_2,a_3$; in particular, the numerator has at most six terms. By way of contrast, it is known that no such expression exists for any $n\geq4$.

On a Two-Sided Turán Problem

10.37236/1735 ◽  
2003 ◽  
Vol 10 (1) ◽  
Author(s):  
Dhruv Mubayi ◽  
Yi Zhao
Keyword(s):  
Minimum Size ◽  
Turan Problem ◽  
Turan Problems ◽  
Turán Problem ◽  
Positive Integers

Given positive integers $n,k,t$, with $2 \le k\le n$, and $t < 2^k$, let $m(n,k,t)$ be the minimum size of a family ${\cal F}$ of nonempty subsets of $[n]$ such that every $k$-set in $[n]$ contains at least $t$ sets from ${\cal F}$, and every $(k-1)$-set in $[n]$ contains at most $t-1$ sets from ${\cal F}$. Sloan et al. determined $m(n, 3, 2)$ and Füredi et al. studied $m(n, 4, t)$ for $t=2, 3$. We consider $m(n, 3, t)$ and $m(n, 4, t)$ for all the remaining values of $t$ and obtain their exact values except for $k=4$ and $t= 6, 7, 11, 12$. For example, we prove that $ m(n, 4, 5) = {n \choose 2}-17$ for $n\ge 160$. The values of $m(n, 4, t)$ for $t=7,11,12$ are determined in terms of well-known (and open) Turán problems for graphs and hypergraphs. We also obtain bounds of $m(n, 4, 6)$ that differ by absolute constants.

A Note on Differential Identities in Prime and Semiprime Rings

2020 ◽  
pp. 77-83
Author(s):  
Mohammad Shadab Khan ◽  
Mohd Arif Raza ◽  
Nadeemur Rehman
Keyword(s):  
Prime Ring ◽  
Semiprime Ring ◽  
Nonzero Ideal ◽  
Differential Identities ◽  
Standard Identity ◽  
Prime And Semiprime Rings ◽  
Semiprime Rings ◽  
Positive Integers

Let R be a prime ring, I a nonzero ideal of R, d a derivation of R and m, n fixed positive integers. (i) If (d ( r ○ s)(r ○ s) + ( r ○ s) d ( r ○ s)n - d ( r ○ s))m for all r, s ϵ I, then R is commutative. (ii) If (d ( r ○ s)( r ○ s) + ( r ○ s) d ( r ○ s)n - d (r ○ s))m ϵ Z(R) for all r, s ϵ I, then R satisfies s4, the standard identity in four variables. Moreover, we also examine the case when R is a semiprime ring.

scholarly journals The Irregularity and Modular Irregularity Strength of Fan Graphs

Symmetry ◽  
2021 ◽  
Vol 13 (4) ◽  
pp. 605
Author(s):  
Martin Bača ◽  
Zuzana Kimáková ◽  
Marcela Lascsáková ◽  
Andrea Semaničová-Feňovčíková
Keyword(s):  
Simple Graph ◽  
Isolated Vertex ◽  
Positive Integers ◽  
Irregularity Strength

For a simple graph G with no isolated edges and at most, one isolated vertex, a labeling φ:E(G)→{1,2,…,k} of positive integers to the edges of G is called irregular if the weights of the vertices, defined as wtφ(v)=∑u∈N(v)φ(uv), are all different. The irregularity strength of a graph G is known as the maximal integer k, minimized over all irregular labelings, and is set to ∞ if no such labeling exists. In this paper, we determine the exact value of the irregularity strength and the modular irregularity strength of fan graphs.

scholarly journals On the Ternary Exponential Diophantine Equation Equating a Perfect Power and Sum of Products of Consecutive Integers

Mathematics ◽  
2021 ◽  
Vol 9 (15) ◽  
pp. 1813
Author(s):  
S. Subburam ◽  
Lewis Nkenyereye ◽  
N. Anbazhagan ◽  
S. Amutha ◽  
M. Kameswari ◽  
...  
Keyword(s):  
Positive Integer ◽  
Diophantine Equation ◽  
Upper Bound ◽  
Exponential Diophantine Equation ◽  
Research Article ◽  
All Solutions ◽  
Sum Of Products ◽  
Consecutive Integers ◽  
Sums Of Products ◽  
Positive Integers

Consider the Diophantine equation yn=x+x(x+1)+⋯+x(x+1)⋯(x+k), where x, y, n, and k are integers. In 2016, a research article, entitled – ’power values of sums of products of consecutive integers’, primarily proved the inequality n= 19,736 to obtain all solutions (x,y,n) of the equation for the fixed positive integers k≤10. In this paper, we improve the bound as n≤ 10,000 for the same case k≤10, and for any fixed general positive integer k, we give an upper bound depending only on k for n.

Sign in / Sign up

Export Citation Format

Share Document