Dense Edge-Magic Graphs and Thin Additive Bases

Pertfect matching and zero-sum 3-magic labeling

A mapping $l : E(G) \rightarrow A$, where $A$ is an abelian group which written additively, is called a labeling of the graph $G$. For every positive integer $h \geqslant 2$, a graph $G$ is said to be zero-sum $h$-magic if there is an edge labeling $l$ from $E(G)$ into $\mathbb{Z}_{h} \backslash \{0\}$ such that $s(v) = \sum_{uv\in E(G)}l(uv) = 0$ for every vertex $v \in V(G)$. In 2014, Saieed Akbari, Farhad Rahmati and Sanaz Zare conjectured that every 5-regular graph admits a zero-sum $3$-magic labeling. In this paper, we obtained that every 5-regular graph with every edge contains in a tria…

Convex sequences may have thin additive bases

For a fixed $c > 0$ we construct an arbitrarily large set $B$ of size $n$ such that its sum set $B+B$ contains a convex sequence of size $cn^2$, answering a question of Hegarty.

群顶点魔法图的谱

Spectra of Group Vertex Magic Graphs

该研究引入图的群魔法谱与约化谱概念，提出基于最少顶点数的新方法，得到特定阿贝尔群下图谱为子群的充要条件，并探究两类谱的关系。

Group distance magic and antimagic graphs

Given a graph $G$ with $n$ vertices and an Abelian group $A$ of order $n$, an $A$-distance antimagic labelling of $G$ is a bijection from $V(G)$ to $A$ such that the vertices of $G$ have pairwise distinct weights, where the weight of a vertex is the sum (under the operation of $A$) of the labels assigned to its neighbours. An {$A$-distance magic labelling} of $G$ is a bijection from $V(G)$ to $A$ such that the weights of all vertices of $G$ are equal to the same element of $A$. In this paper we study these new labellings under a general setting with a focus on product graphs. We prove among ot…

Shatrovskii's construction of thin bases

The set A of nonnegative integers is called a basis of order h if every nonnegative integer can be represented as the sum of exactly h not necessarily distinct elements of A. An additive basis A of order h is called thin if there exists c > 0 such that the number of elements of A not exceeding x is less than cx^{1/h} for all x > 0. This paper describes a construction of Shatrovskii of thin bases of order h.

Cassels bases

This paper describes several classical constructions of thin bases of finite order in additive number theory, and, in particular, gives a complete presentation of a beautiful construction of J. W. S. Cassels of a class of polynomially asymptotic bases. Some open problems are also discussed.