where $\chi(G)$ is the chromatic number of $G$ and $\eta(G)$ is the additive coloring number of $G$, defined below. The problem as well as the conjecture has recently gained interest from the scientific community [@LUCKYCOTAINF; @ORLOW; @LUCIANO; @LUCKYCOMPLEXITY; @LUCKYCOTASUP; @ADDITIVEPLANAR; @LUCKYCHOICE; @DLUCKY]. However, the additive chromatic number is known for very few families of graphs.

Below, we make some basic definitions to formalize these concepts. For a given integer $k$, denote the set $\{1,2,\ldots,k\}$ with $[k]$. Let $G = (V, E)$ be a finite, undirected and simple graph. Usually, $V = [n]$ where $n$ is the number of vertices of $G$. For each $v \in V$, let $N_G(v)$ be the set of neighbors of $v$ and $d_G(v)$ its degree, $d_G(v) = |N_G(v)|$. Also, $N_G[v] = N_G(v) \cup \{v\}$. Let $D = (V, A)$ be a finite directed graph. For each $v \in V$, define $N_D^-(v) = \{(u,v) \in A : u \in V\}$, $N_D^+(v) = \{(v,w) \in A : w \in V\}$ and $N_D(v) = N_D^-(v) \cup N_D^+(v)$. When the graph or digraph is inferred from the context, we omit the subindex, $d(v), N(v), N[v], N^-(v), N^+(v)$.

Let $f : V \rightarrow [k]$ be a labeling of the vertices of 