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Lucky labelings of graphs. (English) Zbl 1197.05125

Summary: Suppose the vertices of a graph $G$ were labeled arbitrarily by positive integers, and let $S\left(v\right)$ denote the sum of labels over all neighbors of vertex $v$. A labeling is lucky if the function $S$ is a proper coloring of $G$, that is, if we have $S\left(u\right)\ne S\left(v\right)$ whenever $u$ and $v$ are adjacent. The least integer $k$ for which a graph $G$ has a lucky labeling from the set $\left\{1,2,\cdots ,k\right\}$ is the lucky number of $G$, denoted by $\eta \left(G\right)$.

Using algebraic methods we prove that $\eta \left(G\right)⩽k+1$ for every bipartite graph $G$ whose edges can be oriented so that the maximum out-degree of a vertex is at most $k$. In particular, we get that $\eta \left(T\right)⩽2$ for every tree $T$, and $\eta \left(G\right)⩽3$ for every bipartite planar graph $G$. By another technique we get a bound for the lucky number in terms of the acyclic chromatic number. This gives in particular that $\eta \left(G\right)⩽100280245065$ for every planar graph $G$. Nevertheless we offer a provocative conjecture that $\eta \left(G\right)⩽\chi \left(G\right)$ for every graph $G$.

##### MSC:
 05C78 Graph labelling