= n$ [@LUCKYORIGINAL].\
When a graph has an additive coloring, an acyclic orientation of this graph arises. In fact, one can obtain the additive chromatic number of a graph by exploring their acyclic orientations and solving, for each one, a problem called *Topological Additive Numbering* (TAN) [@IPL2013]. We introduce more definitions in order to explain this approach. Let $D = (V, A)$ be a directed acyclic graph and $G(D)$ be the undirected underlying graph of $D$. We say that $D$ *represents an acyclic orientation* of $G$ if $G(D)$ is isomorphic to $G$. Let $f: V \rightarrow [k]$ be a labeling of vertices of $D$. If $f(N(u)) < f(N(v))$ for every $(u, v) \in A$, then $f$ is called *topological additive $k$-numbering* of $D$. The *topological additive number* of $D$, denoted by $\eta_t(D)$, is defined as the least number $k$ for which $D$ has a topological additive $k$-numbering, or $+\infty$ in case that such $k$ does not exist (knowing this parameter is -hard [@IPL2013]). Now, the following relationship becomes apparent: $$\eta(G) = \min \{ \eta_t(D) ~:~ D~\textrm{represents an acyclic orientation of}~G \}$$ We can take advantage of properties known for TAN. For instance, the following result provides a lower bound of 