$\eta_t(D)$ and, therefore, $\eta(G)$:

[@IPL2013] Let $D = (V,A)$ be a directed acyclic graph such that its vertices are ordered so that $(u, v) \in A$ implies $u < v$. If $Q$ is a clique of $G(D)$ and $q_F$, $q_L$ are the smallest and largest vertices of $Q$ respectively, then $$\eta_t(D) \geq \biggl\lceil \dfrac{d(q_F)+1}{d(q_L)-|Q|+2} \biggr\rceil.$$

\[COTAINF\] Let $G$ be a graph and $Q$ be a clique of $G$. If $d_1$, $d_2$ are the degrees of the vertices of $Q$ with smallest and largest degree respectively, then $$\eta(G) \geq \biggl\lceil \dfrac{d_1+1}{d_2-|Q|+2} \biggr\rceil.$$

The latter bound can be relaxed by considering $d_1 \geq |Q|-1$ and $d_2 \leq n-1$. Hence, $\eta(G) \geq \lceil \frac{|Q|}{n-|Q|+1} \rceil$, which is a lower bound previously proposed in [@LUCKYCOTAINF].\
As we mentioned before, one of the reasons to study ACP is that this problem and GCP seem to be related as follows:

**Additive Coloring Conjecture.** [@LUCKYORIGINAL] For every graph $G$, $\eta(G) \leq \chi(G)$.

It is known that the conjecture holds for trees [@LUCKYORIGINAL; @ORLOW] and, recently, for non-bipartite planar graphs of girth at least 26 [@LUCKYCHOICE]. Our contribution in this work is to give the exact value of the additive chromatic number of several families of graphs and 