The inapproximability for the (0,1)-additive number

Abstract

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 $\eta(G)$, 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 $\eta_{\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 $G$, $\eta(G)= \eta_{\ell}(G)$. We give a negative answer to this conjecture and we show that for every $k$ there is a graph $G$ such that $ \eta_{\ell}(G)- \eta(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 $ \sigma_{1} (G) = \min_{\ell \in \Gamma}\sum_{v\in V(G)}\ell(v) $ where $\Gamma$ 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.