On the Lucky labeling of Graphs

Suppose the vertices of a graph $G$ were labeled arbitrarily by positive integers, and let $Sum(v)$ denote the sum of labels over all neighbors of vertex $v$. A labeling is lucky if the function $Sum$ is a proper coloring of $G$, that is, if we have $Sum(u) \neq Sum(v)$ whenever $u$ and $v$ are adjacent. The least integer $k$ for which a graph $G$ has a lucky labeling from the set $\lbrace 1, 2, ...,k\rbrace$ is the lucky number of $G$, denoted by $\eta(G)$. We will prove, for every graph $G$ other than $K_{2}$, $\frac{w}{n-w+1}\leq\eta (G) \leq \Delta^{2}$ and we present an algorithm for lucky labeling of $G$.