other hand, if $G'$ is a subgraph of $G$, we have $\chi(G') \leq \chi(G)$ but the same property does not hold for ACP. For instance, $\eta(P_2) = 2$ but $\eta(P_3) = 1$. And, for graphs $G$ with maximum degree $\Delta$, the best known upper bound of $\eta(G)$ is $\Delta^2 - \Delta + 1$ [@LUCKYCOTASUP], as opposed to Brooks’ result for GCP ($\chi(G) \leq \Delta+1$) which is significantly better.

Constant upper bounds of the additive chromatic number are known for some families of graphs: if $G$ is a tree, $\eta(G) \leq 2$; if $G$ is planar bipartite, $\eta(G) \leq 3$ [@LUCKYORIGINAL]; if $G$ is planar, $\eta(G) \leq 468$ [@ADDITIVEPLANAR] and if $G$ is planar of girth at least 26, $\eta(G) \leq 3$ [@LUCKYCHOICE]. Other upper bounds can be consulted in [@LUCKYORIGINAL; @ADDITIVEPLANAR; @LUCKYCHOICE].

Regarding lower bounds of $\eta(G)$, one of them can be computed as follows. If $G$ has *true twins* vertices $u$ and $v$ ($N[u] = N[v]$), then an additive coloring $f$ of $G$ must satisfy $f(u) \neq f(v)$. Therefore:

\[LOWERBOUNDTWINS\] Let $T \subset V$ such that any $u,v \in T$ are true twins of $G$. Then, $\eta(G) \geq |T|$.

The given formula can be applied to prove that $\eta(K_n) 