of $G$ that provides the lowest value of $\eta_t(D)$. Let $D'$ be another digraph representing an acyclic orientation of $G$ with $\eta_t(D') < \infty$. Therefore, $D'$ is a monotone complete $r$-partite digraph where $G(D')$ is isomorphic to $G$ and the partition of $V(D')$ is $V'_i = V_{\mathfrak{p}(i)}$ for all $i \in [r]$ where $\mathfrak{p}:[r] \rightarrow [r]$ is some permutation function. Define $s_i$ and $s'_i$ for $D$ and $D'$ respectively as in Lemma \[LEMITA\]. It is easy to verify that sequences $\{s_i\}_{i \in [r]}$ and $\{s'_i\}_{i \in [r]}$ are decreasing, and $s'_i \geq s_i$ for all $i \in [r]$. Let $i$ be an integer such that $s_i/|V_i|$ is maximum and $I = \{ t \in [r] : |V_t| = |V_i|\}$. Note that $i$ is the minimum index of $I$. Let $J = \{ t \in [r] : |V'_t| = |V_i|\}$ and $j$ be the minimum index of $J$. Due to the ordering in the cardinality of sets of $V(D)$, $i \geq j$. Hence, $s'_j \geq s'_i \geq s_i$. Since $j \in J$, $|V'_j| = |V_i|$ and we obtain $s'_j/|V'_j| \geq s_i/|V_i|$. Therefore, $\eta_t(D') \geq \lceil s'_j/|V'_j| \rceil \geq \lceil s_i/|V_i| \rceil = \eta_t(D)$.

Now, we show that $\eta(G) \leq r$. We 