Related papers: Restricted-sum-dominant sets

Let $G$ be an additive abelian group. A sequence $S = g_1 \cdot \ldots \cdot g_{\ell}$ of terms from $G$ is a plus-minus weighted zero-sum sequence if there are $\varepsilon_1, \ldots, \varepsilon_{\ell} \in \{-1, 1\}$ such that…

Let $A=\{a_0,a_1,\ldots,a_{k-1}\}$ be a set of $k$ integers. For any integer $h\ge 1$ and any ordered $k$-tuple of positive integers $\mathbf{r}=(r_0,r_1,\ldots,r_{k-1})$, we define a general $h$-fold sumset, denoted by $h^{(\mathbf{r})}A$,…

We study the structure of infinite discrete sets D definable in expansions of ordered Abelian groups whose theories are strong and definably complete, with particular emphasis on the set D' comprised of differences between successive…

A dominating set in a graph $G$ is a set $S$ of vertices such that every vertex in $V(G) \setminus S$ is adjacent to a vertex in $S$. A restrained dominating set of $G$ is a dominating set $S$ with the additional restraint that the graph $G…

Let $G$ be a finite abelian group. The critical number ${\rm cr}(G)$ of $G$ is the least positive integer $\ell$ such that every subset $A\subseteq G\setminus\{0\}$ of cardinality at least $\ell$ spans $G$, i.e., every element of $G$ can be…

Let $A$ be a finite set of $k$ integers. For $h \leq k$, the restricted $h$-fold sumset $h^{\wedge} A$ is the set of all sums of $h$ distinct elements of $A$. In additive combinatorics, much of the focus has traditionally been on finite…

In this note, we use integral binary cubic forms to study the rational cube sum problem. We prove (unconditionally) that for any positive integer $d$, infinitely many primes in each of the residue classes $ 1 \pmod {9d}$ as well as $ -1…

A dominating set of a graph $G=(V,E)$ is a vertex set $D$ such that every vertex in $V(G) \setminus D$ is adjacent to a vertex in $D$. The cardinality of a smallest dominating set of $D$ is called the domination number of $G$ and is denoted…

A numerical set is a co-finite subset of the natural numbers that contains zero. Its Frobenius number is the largest number in its complement. Each numerical set has an associated semigroup $A(T)=\{t\mid t+T\subseteq T\}$, which has the…

For every positive integer $k$, a set $S$ of vertices in a graph $G=(V,E)$ is a $k$-tuple dominating set of $G$ if every vertex of $V-S$ is adjacent to least $k$ vertices and every vertex of $S$ is adjacent to least $k-1$ vertices in $S$.…

The domination polynomial D(G,x) is the ordinary generating function for the dominating sets of an undirected graph G=(V,E) with respect to their cardinality. We consider in this paper representations of D(G,x) as a sum over subsets of the…

We extend the concepts of sum-free sets and Sidon-sets of combinatorial number theory with the aim to provide explicit constructions for spherical designs. We call a subset $S$ of the (additive) abelian group $G$ {\it $t$-free} if for all…

Let $A$ and $B$ be additive sets of $\mathbb{Z}_{2k}$, where $A$ has cardinality $k$ and $B=v.\complement A$ with $v\in\mathbb{Z}_{2k}^{\times}$. In this note some bounds for the cardinality of $A+B$ are obtained, using four different…

The purpose of the article is to provide an unified way to formulate zero-sum invariants. Let $G$ be a finite additive abelian group. Let $B(G)$ denote the set consisting of all nonempty zero-sum sequences over G. For $\Omega \subset B(G$),…

A group $G$ is said to have restricted centralizers if for each $g \in G$ the centralizer $C_G(g)$ either is finite or has finite index in $G$. Shalev showed that a profinite group with restricted centralizers is virtually abelian. We take…

The concept of majority domination in graphs has been defined in at least two different ways: As a function and as a set. In this work we extend the latter concept to digraphs, while the former was extended in another paper. Given a digraph…

A set of points $S$ in Euclidean space $\mathbb{R}^d$ is called \textit{Ramsey} if any finite partition of $\mathbb{R}^{\infty}$ yields a monochromatic copy of $S$. While characterization of Ramsey set remains a major open problem in the…

Let $\mathrm{d}(A)$ be the asymptotic density (if it exists) of a sequence of integers $A$. For any real numbers $0\leq\alpha\leq\beta\leq 1$, we solve the question of the existence of a sequence $A$ of positive integers such that…

We consider restricted sumsets over field $F$. Let\begin{align*}C=\{a_1+\cdots+a_n:a_1\in A_1,\ldots,a_n\in A_n, a_i-a_j\notin S_{ij}\ \text{if}\ i\not=j\},\end{align*} where $S_{ij}(1\leqslant i\not=j\leqslant n)$ are finite subsets of $F$…

Let $\mathcal{A}$ be a sequence of $rk$ terms which is made up of $k$ distinct integers each appearing exactly $r$ times in $\mathcal{A}$. The sum of all terms of a subsequence of $\mathcal{A}$ is called a subsequence sum of $\mathcal{A}$.…