Intersective sets over abelian groups

We prove that the sizes of the maximal dissociated subsets of a given finite subset of an abelian group differ by a logarithmic factor at most. On the other hand, we show that the set $\{0,1\}^n\seq\Z^n$ possesses a dissociated subset of size $\Ome(n\log n)$; since the standard basis of $\Z^n$ is a maximal dissociated subset of $\{0,1\}^n$ of size $n$, the result just mentioned is essentially sharp.

Let G be a group. The intersection graph G(G) of G is an undirected graph without loops and multiple edges defined as follows: the vertex set is the set of all proper nontrivial subgroups of G; and there is an edge between two distinct vertices X and Y if and only if X \ Y 6= 1 where 1 denotes the trivial subgroup of G: It was conjectured in [2] that two (non cyclic) finite abelian groups with isomorphic intersection graphs are isomorphic. In this paper we study this conjectu...

Let $G$ be a finite group. For maximal cyclic subgroups $M, N$ of $G$, we denote by $\text{d}(M,N)$ the minimum of number of elements in the set differences $M {\setminus} N$ and $N {\setminus} M$. The difference number $\delta(G)$ of $G$ is defined as the maximum of $\text{d}(M,N)$ as $M$ and $N$ vary over every pair of maximal cyclic subgroups of $G$. Whereas, the power graph $\Gamma_G$ of $G$ is the undirected simple graph with vertex set $G$ and two distinct vertices are ...

In 1965, Erd\H{o}s and P\'{o}sa proved that there is a duality between the maximum size of a packing of cycles and the minimum size of a vertex set hitting all cycles. Such a duality does not hold for odd cycles, and Dejter and Neumann-Lara asked in 1988 to find all pairs ${(\ell, z)}$ of integers where such a duality holds for the family of cycles of length $\ell$ modulo $z$. We characterise all such pairs, and we further generalise this characterisation to cycles in graphs ...

Let G be a finite additive abelian group with exponent exp(G)=n>1 and let A be a nonempty subset of {1,...,n-1}. In this paper, we investigate the smallest positive integer $m$, denoted by s_A(G), such that any sequence {c_i}_{i=1}^m with terms from G has a length n=exp(G) subsequence {c_{i_j}}_{j=1}^n for which there are a_1,...,a_n in A such that sum_{j=1}^na_ic_{i_j}=0. When G is a p-group, A contains no multiples of p and any two distinct elements of A are incongruent m...

For a finite Abelian group $(\Gamma,+)$, let $n(\Gamma)$ denote the smallest positive integer $n$ such that for each labelling of the arcs of the complete digraph of order $n$ using elements from $\Gamma$, there exists a directed cycle such that the total sum of the arc-labels along the cycle equals $0$. Alon and Krivelevich initiated the study of the parameter $n(\cdot)$ on cyclic groups and proved that $n(\mathbb{Z}_q)=O(q\log q)$. Studying the prototypical case when $\Gamm...

Suppose $G$ is a finite abelian group and $S$ is a sequence of elements in $G$. For any element $g$ of $G$, let $N_g(S)$ denote the number of subsequences of $S$ with sum $g$. The purpose of this paper is to investigate the lower bound for $N_g(S)$. In particular, we prove that either $N_g(S)=0$ or $N_g(S) \ge 2^{|S|-D(G)+1}$, where $D(G)$ is the smallest positive integer $\ell$ such that every sequence over $G$ of length at least $\ell$ has a nonempty zero-sum subsequence. W...

Let $G$ be a group. The intersection graph $\Gamma(G)$ of $G$ is an undirected graph without loops and multiple edges defined as follows: the vertex set is the set of all proper non-trivial subgroups of $G$, and there is an edge between two distinct vertices $H$ and $K$ if and only if $H\cap K \neq 1$ where $1$ denotes the trivial subgroup of $G$. In this paper we studied the dominating sets in intersection graphs of finite groups. It turns out a subset of the vertex set is a...

Let $G$ be a finite abelian group of order $n$. For any subset $B$ of $G$ with $B=-B$, the Cayley graph $G_B$ is a graph on vertex set $G$ in which $ij$ is an edge if and only if $i-j\in B.$ It was shown by Ben Green that when $G$ is a vector space over a finite field $Z/pZ$, then there is a Cayley graph containing neither a complete subgraph nor an independent set of size more than $clog nloglog n,$ where $c$ is an absolute constant. In this article we observe that a modific...

Let $A$ be a subset of the cyclic group $\mathbf{Z}/p\mathbf{Z}$ with $p$ prime. It is a well-studied problem to determine how small $|A|$ can be if there is no unique sum in $A+A$, meaning that for every two elements $a_1,a_2\in A$, there exist $a_1',a_2'\in A$ such that $a_1+a_2=a_1'+a_2'$ and $\{a_1,a_2\}\neq \{a_1',a_2'\}$. Let $m(p)$ be the size of a smallest subset of $\mathbf{Z}/p\mathbf{Z}$ with no unique sum. The previous best known bounds are $\log p \ll m(p)\ll \sq...