图$G$的{ \it加法标号}是函数$ \ell :V(G) \rightarrow\mathbb{N}$，使得对于$ G $的每两个相邻顶点$ v $和$ u$，$ \sum_{w \sim v}\ell(w)\neq \sum_{w \sim u}\ell(w) $（$ x \sim y $意味着$ x $连接到$y$）。由$η(G)$表示的$ G $的{ \it加法数}是使得$ G $具有加法标记$ \ell :V(G) \rightarrow \mathbb{N}_k$的最小数$k $。图$G$的{ \it加性选择性}，记为$η_{\ell}(G) $，是最小数$k$，使得$G$对于$G$的顶点的大小为$k$的列表的任何分配具有加性标记，使得每个顶点的标记属于其自己的列表。\cite{a80}（2012）证明了对于每个图$G$，$η(G)= η_{\ell}(G)$。我们给出了一个否定的答案，这个猜想，我们表明，对于每一个$k$有一个图$G$，使$ η_{\ell}(G)- η(G) \geq k$。图$G$的{ \it $(0,1)$ -加法标号}是函数$ \ell :V(G) \rightarrow\{0,1\}$，使得对于$ G $的每两个相邻顶点$ v $和$ u$，$ \sum_{w \sim v}\ell(w)\neq \sum_{w \sim u}\ell(w) $。图表可能缺少任何$(0,1)$添加剂标记。我们证明了对于某些图族，如完美图和平面无三角形图，判定是否存在$(0,1)$ -可加标号是$ \mathbf{NP} $ -完全的。对于具有一些$(0,1)$ -可加标号的图$G$，$G$的$(0,1)$ -可加数被定义为$ σ_{1} (G) = \min_{\ell \in Γ}\sum_{v\in V(G)}\ell(v) $，其中$Γ$是$G$的$(0,1)$ -可加标号的集合。我们证明了给定一个平面图，允许一个$(0,1)$ -加性标签，对于所有的$ \varepsilon >0 $，逼近的$(0,1)$ -加性数内$ n^{1-\varepsilon} $是$ \mathbf{NP} $ -硬。
英文摘要：
An
{\it additive labeling} of a graph $G$ is a function $ \ell :V(G) \rightarrow\mathbb{N}$, such that for every two adjacent vertices $ v $ and $ u$ of $ G $, $ \sum_{w \sim v}\ell(w)\neq \sum_{w \sim u}\ell(w) $ ($ x \sim y $ means that $ x $ is joined to $y$). The {\it additive number} of $ G $, denoted by $η(G)$, is the minimum number $k $ such that $ G $ has a additive labeling $ \ell :V(G) \rightarrow \mathbb{N}_k$. The {\it additive choosability} of a graph $G$, denoted by $η_{\ell}(G) $, is the smallest number $k$ such that $G$ has an additive labeling for any assignment of lists of size $k$ to the vertices of $G$, such that the label of each vertex belongs to its own list.
Seamone (2012) \cite{a80} conjectured that for every graph $G$, $η(G)= η_{\ell}(G)$. We give a negative answer to this conjecture and we show that for every $k$ there is a graph $G$ such that $ η_{\ell}(G)- η(G) \geq k$.
A {\it $(0,1)$-additive labeling} of a graph $G$ is a function $ \ell :V(G) \rightarrow\{0,1\}$, such that for every two adjacent vertices $ v $ and $ u$ of $ G $, $ \sum_{w \sim v}\ell(w)\neq \sum_{w \sim u}\ell(w) $.
A graph may lack any $(0,1)$-additive labeling. We show that it is $ \mathbf{NP} $-complete to decide whether a $(0,1)$-additive labeling exists for some families of graphs such as perfect graphs and planar triangle-free graphs. For a graph $G$ with some $(0,1)$-additive labelings, the $(0,1)$-additive number of $G$ is defined as $ σ_{1} (G) = \min_{\ell \in Γ}\sum_{v\in V(G)}\ell(v) $ where $Γ$ is the set of $(0,1)$-additive labelings of $G$. We prove that given a planar graph that admits a $(0,1)$-additive labeling, for all $ \varepsilon >0 $, approximating the $(0,1)$-additive number within $ n^{1-\varepsilon} $ is $ \mathbf{NP} $-hard.
我们研究s（k，n），A+A的最大尺寸，其中A是[n]的k-子集。加性数论中的一些已知函数可以用s（k，n）表示。例如，我们对s（k，n）的估计意味着拟Sidon集的最大大小的新界限，这是Erdos和Freud [J. Number Th.38（1991）196-205]提出的问题。此外，应用所谓的边魔术标号的图。
