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title: On the additive chromatic number of several families of graphs
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Introduction
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Several combinatorial optimization problems concern finding means to distinguish the vertices of a graph. Such identification can be *global*, when each vertex is uniquely identified from the solution of the optimization problem, or *local*, when for every edge $(u,v)$, $u$ and $v$ can be distinguished each other from the solution of the optimization problem. Usually the solution restricted to the closed neighborhood of a vertex is used for that identification, although open neighborhood can be used as well. Most of these problems are coloring problems. On the side of global identification problems we can mention *Identification Code Problem* [@KARPOVSKY] and *Recognizable Coloring of Graphs* [@RECOGNIZABLE]. On the side of local ones, *Locally Identifying Coloring of Graphs* [@ESPERET] and several problems where open neighborhood is used for identification: *Vertex Coloring by Sums*, *Products* and *Multisets* among others [@SEAMONE].

In this paper we address one of these problems, specifically the Vertex Coloring by Sums, which is also called *Additive Coloring Problem* or *Lucky Labeling Problem*. It was first presented by Czerwiński, Grytczuk and Zelazny [@LUCKYORIGINAL] who proposed a conjecture that for every graph $G$, $\eta(G) \leq \chi(G)$, 