scholarly journals Some Constructions of General Covering Designs

10.37236/2606 ◽  
2012 ◽  
Vol 19 (3) ◽  
Author(s):  
Federico Montecalvo
Keyword(s):  
Upper Bounds ◽  
Minimum Size ◽  
Covering Designs ◽  
Covering Design ◽  
Positive Integers

Given five positive integers $v, m,k,\lambda$ and $t$ where $v\geq k \geq t$ and $v \geq m \geq t,$ a $t$-$(v,k,m,\lambda)$ general covering design is a pair $(X,\mathcal{B})$ where $X$ is a set of $v$ elements (called points) and $\mathcal{B}$ a multiset of $k$-subsets of $X$ (called blocks) such that every $m$-subset of $X$ intersects (is covered by) at least $\lambda$ members of $\mathcal{B}$ in at least $t$ points. In this article we present new constructions for general covering designs and we generalize some others. By means of these constructions we will be able to obtain some new upper bounds on the minimum size of such designs.

scholarly journals On $q$-Covering Designs

10.37236/8718 ◽  
2020 ◽  
Vol 27 (1) ◽  
Author(s):  
Francesco Pavese
Keyword(s):  
Upper Bounds ◽  
Minimum Number ◽  
Covering Designs ◽  
Covering Design

A $q$-covering design $\mathbb{C}_q (n, k, r)$, $k \ge r$, is a collection $\mathcal{X}$ of $(k-1)$-spaces of $PG(n-1, q)$ such that every $(r-1)$-space of $PG(n-1,q)$ is contained in at least one element of $\mathcal{X}$ . Let $\mathcal{C}_q(n, k, r)$ denote the minimum number of $(k-1)$-spaces in a $q$-covering design $\mathbb{C}_q (n, k, r)$. In this paper improved upper bounds on $\mathcal{C}_q(2n, 3, 2)$, $n \ge 4$, $\mathcal{C}_q(3n + 8, 4, 2)$, $n \ge 0$, and $\mathcal{C}_q(2n,4,3)$, $n \ge 4$, are presented. The results are achieved by constructing the related $q$-covering designs.

Infinite Classes of Covering Numbers

2000 ◽  
Vol 43 (4) ◽  
pp. 385-396 ◽  
Author(s):  
I. Bluskov ◽  
M. Greig ◽  
K. Heinrich
Keyword(s):  
Upper Bounds ◽  
Covering Number ◽  
Minimum Size ◽  
Covering Numbers ◽  
Covering Design

AbstractLet D be a family of k-subsets (called blocks) of a v-set X(v). Then D is a (v, k, t) covering design or covering if every t-subset of X(v) is contained in at least one block of D. The number of blocks is the size of the covering, and the minimum size of the covering is called the covering number. In this paper we consider the case t = 2, and find several infinite classes of covering numbers. We also give upper bounds on other classes of covering numbers.

scholarly journals General Upper Bounds on the Minimum Size of Covering Designs

1999 ◽  
Vol 86 (2) ◽  
pp. 205-213
Author(s):  
Iliya Bluskov ◽  
Katherine Heinrich
Keyword(s):  
Upper Bounds ◽  
Minimum Size ◽  
Covering Designs

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.

scholarly journals The Minimum Size of Signed Sumsets

10.37236/4881 ◽  
2015 ◽  
Vol 22 (2) ◽  
Author(s):  
Béla Bajnok ◽  
Ryan Matzke
Keyword(s):  
Abelian Group ◽  
Upper Bound ◽  
Finite Abelian Group ◽  
Minimum Size ◽  
Positive Integers

For a finite abelian group $G$ and positive integers $m$ and $h$, we let $$\rho(G, m, h) = \min \{ |hA| \; : \; A \subseteq G, |A|=m\}$$ and$$\rho_{\pm} (G, m, h) = \min \{ |h_{\pm} A| \; : \; A \subseteq G, |A|=m\},$$ where $hA$ and $h_{\pm} A$ denote the $h$-fold sumset and the $h$-fold signed sumset of $A$, respectively. The study of $\rho(G, m, h)$ has a 200-year-old history and is now known for all $G$, $m$, and $h$. Here we prove that $\rho_{\pm}(G, m, h)$ equals $\rho (G, m, h)$ when $G$ is cyclic, and establish an upper bound for $\rho_{\pm} (G, m, h)$ that we believe gives the exact value for all $G$, $m$, and $h$.

scholarly journals Parity of the partition function p(n,k)

2019 ◽  
Vol 15 (04) ◽  
pp. 799-805
Author(s):  
Kedar Karhadkar
Keyword(s):  
Partition Function ◽  
Upper Bounds ◽  
Lower And Upper Bounds ◽  
Positive Integers

Let [Formula: see text] denote the number of partitions of [Formula: see text] into parts less than or equal to [Formula: see text]. We show several properties of this function modulo 2. First, we prove that for fixed positive integers [Formula: see text] and [Formula: see text], [Formula: see text] is periodic modulo [Formula: see text]. Using this, we are able to find lower and upper bounds for the number of odd values of the function for a fixed [Formula: see text].

scholarly journals Symmetry Breaking in Tournaments

10.37236/3182 ◽  
2013 ◽  
Vol 20 (1) ◽  
Author(s):  
Antoni Lozano
Keyword(s):  
Symmetry Breaking ◽  
Upper Bounds ◽  
Minimum Size ◽  
Metric Dimension ◽  
Resolving Set ◽  
Nontrivial Automorphism ◽  
Determining Set ◽  
Strong Tournament

We provide upper bounds for the determining number and the metric dimension of tournaments. A set of vertices $S \subseteq V(T)$ is a determining set for a tournament $T$ if every nontrivial automorphism of $T$ moves at least one vertex of $S$, while $S$ is a resolving set for $T$ if every two distinct vertices in $T$ have different distances to some vertex in $S$. We show that the minimum size of a determining set for an order $n$ tournament (its determining number) is bounded by $\lfloor n/3 \rfloor$, while the minimum size of a resolving set for an order $n$ strong tournament (its metric dimension) is bounded by $\lfloor n/2 \rfloor$. Both bounds are optimal.

scholarly journals Upper Bounds on the Minimum Size of Hamilton Saturated Hypergraphs

10.37236/5252 ◽  
2016 ◽  
Vol 23 (4) ◽  
Author(s):  
Andrzej Ruciński ◽  
Andrzej Żak
Keyword(s):  
Upper Bound ◽  
Upper Bounds ◽  
Minimum Size ◽  
Uniform Hypergraph ◽  
Open Case ◽  
Vertex Ordering

For $1\leqslant \ell< k$,  an $\ell$-overlapping $k$-cycle is a $k$-uniform hypergraph in which, for some cyclic vertex ordering, every edge consists of $k$ consecutive vertices and every two consecutive edges share exactly $\ell$ vertices.A $k$-uniform hypergraph $H$ is $\ell$-Hamiltonian saturated if $H$ does not contain an $\ell$-overlapping Hamiltonian $k$-cycle but every hypergraph obtained from $H$ by adding one edge does contain such a cycle. Let $\mathrm{sat}(n,k,\ell)$ be the smallest number of edges in an $\ell$-Hamiltonian saturated $k$-uniform hypergraph on $n$ vertices. In the case of graphs Clark and Entringer showed in 1983 that $\mathrm{sat}(n,2,1)=\lceil \tfrac{3n}2\rceil$. The present authors proved that for $k\geqslant 3$ and $\ell=1$, as well as for all $0.8k\leqslant \ell\leq k-1$, $\mathrm{sat}(n,k,\ell)=\Theta(n^{\ell})$. In this paper we prove two upper bounds which cover the remaining range of $\ell$. The first, quite technical one, restricted to $\ell\geqslant\frac{k+1}2$, implies in particular that for $\ell=\tfrac23k$ and $\ell=\tfrac34k$ we have $\mathrm{sat}(n,k,\ell)=O(n^{\ell+1})$. Our main result provides an upper bound $\mathrm{sat}(n,k,\ell)=O(n^{\frac{k+\ell}2})$ valid for all $k$ and $\ell$. In the smallest open case we improve it further to $\mathrm{sat}(n,4,2)=O(n^{\frac{14}5})$.

scholarly journals On the $P_3$-Hull Number of Kneser Graphs

10.37236/9903 ◽  
2021 ◽  
Vol 28 (3) ◽  
Author(s):  
Luciano N. Grippo ◽  
Adrián Pastine ◽  
Pablo Torres ◽  
Mario Valencia-Pabon ◽  
Juan C. Vera
Keyword(s):  
Upper Bounds ◽  
Minimum Size ◽  
Lower And Upper Bounds ◽  
Kneser Graph ◽  
Graph Homomorphisms ◽  
The Family ◽  
Vertex Set ◽  
Kneser Graphs

This paper considers an infection spreading in a graph; a vertex gets infected if at least two of its neighbors are infected. The $P_3$-hull number is the minimum size of a vertex set that eventually infects the whole graph. In the specific case of the Kneser graph $K(n,k)$, with $n\ge 2k+1$, an infection spreading on the family of $k$-sets of an $n$-set is considered. A set is infected whenever two sets disjoint from it are infected. We compute the exact value of the $P_3$-hull number of $K(n,k)$ for $n>2k+1$. For $n = 2k+1$, using graph homomorphisms from the Knesser graph to the Hypercube, we give lower and upper bounds.

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